What is Complementary Logic

[Organic]

Please look at:

http://www.geocities.com/complementarytheory/BFC.pdf

Please reply your remarks and insights.

Thank you.

Organic

[master_coda]

Well, the biggest problem seems to be that you don't actually provide any kind of logic system. You just state the results you think your logic can provide.

This has been a general theme of your posts on complimentary logic. You continue to make statements like "Complimentary Logic can be used to do this". Until we see what comp logic is, we don't know what it can do. And all your attempts do define it so far have involved you invoking other undefined terms.

"associations between opposite concepts" may be a valid statement in English, but it is not a valid mathematical one. The English meaning of this sentance is too vague and nebulous. "Opposite" in particular is difficult to define mathematically. And you can't use existing math to define your concepts either, since math is based on logic you don't want to use.


Also, I should point out that a form of logic with no contradictions doesn't seem to be particularly useful. Contradictions help us determine which of our assumptions are good, and which are bad. If we get a contradiction, it means we made a bad assumption.

Thus it seems to me that if a logical system is unable to derive contradictions from bad assumptions, then it's doubtful it can derive tautologies from good assumptions. And that's kind of the whole point of logic, at least as far as math is concerned.

[Organic]

Hi master_coda,

If we find efficient methods to explore and use a vary complex structural and dynamical real time phenomena, then very powerful models can become an actual realty.

This is a very dangerous state if the gap between our moral level and our technological level is too big.

Therefore I think that powerful language like Mathematics has to include the developer|development relations as natural part of it.

Maybe CL point of view is in the right way to this goal, because it first of all based on the idea of associations between opposite things, that naturally destroying each other by association.

If we learn how to develop methods that can help us to communicate with each other in non-distractive ways, I think maybe we will be able to survive the power of future’s technology.

More than that:

I think one of the beautiful things in open systems is not to find THE SOLUTION to something, but to find solutions by active participation through non-destructive communication.

By this attitude i think we always tuning ourselves in real time.

Because Complementary Logic is an open system by nature (and by saying this I mean that any of its results is under the lows of probability that can be clearly shown here:
http://www.geocities.com/complement...ry/Identity.pdf ) then any moment is the time to choose if we survive our choices or not.

Moral conclusions based on CL can be found here:

http://www.geocities.com/complementarytheory/Moral.pdf

Shortly speaking, any decision is based on our abilities to destroy or construct our system.

The answer to this dilemma cannot fully found in the limits of the examined system simply because of the fact that we always can destroy what we can explore.

CL solution to this dilemma is real time participation through communication between opposite things, by discovering their abilities to complement each other to something which its quality is more valuable then any one of them alone.



I think that not be aware to the potential destructive power of mathematics language, by disconnect it from the real world complexity, this is the big problem of Boolean-Logic, and nobody else but us is going to pay for it.



What do you think?

(Please be aware that Boolean Logic and Fuzzy Logic are private cases of CL).

[master_coda]

None of what you posted is relevant, because you still haven't addressed my main point, which is that you haven't actually provided us with a system of logic yet. All of your posts have been assertions about what you can do with your logic, and why you think it's better than traditional logic. But without the actual logic system, those claims don't have any relevance.

If you came here claiming to have a design for a spaceship that could travel faster than light, but refused to give a detailed or precise explaination of how it worked, then it wouldn't matter how important you thought your discovery was, or what you thought you could do with it.


I'm also more convinced than ever that your logic has nothing to do with math. You seem unable to grasp that concept that math has nothing to do with the real world by design. That isn't because the real world is unimportant or irrelevant, but because there are already other fields in philosophy and science that deal with the real world.

If you want to study how logic relates to the real world, go study the fields in philosophy that deal with how logic relates to the real world.

[Organic]

master_coda,

First, BL and ZL are private cases in CL:
http://www.geocities.com/complementarytheory/BFinC.pdf

But CL has the built-in ability to deal with non-linearity, by connecting in a coherent way concepts like symmetry-degree that related to information’s clarity-degree.

Please look at: http://www.geocities.com/complementarytheory/ET.pdf
where I construct the natural number by complementary associations between its integral side (its sum) and its differential side (some finite collection of 1’s).

Through this attitude addition and multiplication are complementary operations.

Show me how can you do that by using Boolean Logic.

[master_coda]

Originally posted by Organic
master_coda,

First, BL and ZL are private cases in CL.

But CL has the built-in ability to deal with non-linearity, by connecting in a coherent way concepts like symmetry-degree that related to information’s clarity-degree.

Please look at: http://www.geocities.com/complementarytheory/ET.pdf
where I construct the natural number by complementary associations between its integral side (its sum) and its differential side (some finite collection of 1’s).

Through this attitude addition and multiplication are complementary operations.

Show me how can you do that by using Boolean Logic.

It can deal with non-linearity? Wow. And all these years I've been wasting my time with traditional logic. Of course, since traditional logic can handle non-linearity, wasted is probably the wrong word. Unless by non-linearity you mean "something I made up and I'm not telling you the definition of".


You also mention that you can construct the natural numbers with CT. But your construction has non-commutative multiplication. So it can't be the natural numbers, because the natural numbers have commutative multiplication.

Bragging about a construction of N with non-commutative multiplication is like bragging about building a car with no brakes. A car is supposed to have brakes. And the natural numbers are supposed to have commutative multiplication. So your construction has failed to accomplish its goals.


As for addition and multiplication being "complimentary", I don't really see how that is so. You've managed to make up other operations and have said they're complimentary, but those operations have little to do with any existing definitions of addition and multiplication.

And you've hardly shown why your goals are worthwhile. All you've done is make up a goal that you think math can't accomplish and said "that should be the goal of math". The fact that math fails to accomplish vague goals that you've made up on the spot is hardly remarkable. Pick any field of science or philosophy, and I can make up a goal that the field has failed at.


Finally, my point still remains...you have yet to do anything but make assertions about what CL can do. Saying "BL and ZL are private cases in CL" doesn't mean anything until you tell us what CL is, and stop telling us about all the wonderful things you think CL will accomplish.

And note: a picture is not a definition. A picture may "be worth a 1000 words", but those words are usually vague and ambiguous. Simply churning out pages and pages of assertions and pictures is something anybody can do if they have a lot of time on their hands. It's more impressive if you can produce a single page of precise, unambiguous content.

[Organic]

master_coda,

You don't read to understand what I write for example:

I wrote:

"But CL has the built-in ability to deal with non-linearity, by connecting in a coherent way concepts like symmetry-degree that related to information’s clarity-degree."

You cut the first line of it and then run to show me how smart you are.

If you read all of it then you will see that I am talking about non-linearity which is based on the ability to connent in a coherent way concepts like symmetry-degree that related to information’s clarity-degree.

If you look at:
http://www.geocities.com/complementarytheory/ET.pdf

and try to find your definitions there in your terms, then don't waste your time.

But if you want to understand someone's point of view then please ask specific questions about something that you don’t understand in ET.pdf and please stop giving me general lecturers about what is Math how Math should be and so on.

About non-commutative and commutative multiplication, let me tell you a secret, commutative multiplication is a private case of non-commutative multiplication because of this reason:

Commutative point of view can see only the quantitative property of the natural number, but the non-commutative point of view can see both structural and quantitative properties of the natural number.

Also what you call natural number is only the private case of an ET with no redundancy and no uncertainty.

For me the Natural number is what I call ET.

My pictures in ET.pdf are precise and rigorous exactly like any mathematical definition.

If you don't think so it means that you understand my pictures and in this case take one of my pictures and show me why it is not precise and rigorous as mathematical definition should be.

Thank you.

Organic

[master_coda]

Originally posted by Organic
But if you want to understand someone's point of view then please ask specific questions about something that you don’t understand in ET.pdf and please stop giving me general lecturers about what is Math how Math should be and so on.

You've been asked many specific questions in the past. You've been asked to define terms without using more undefined terms. You've been asked to prove your assertions. You've been asked to provide the rules of the your "complimentary logic". And you've done none of these things.

As for telling you how math is...if you're going to post stuff which is not math in a math forum, you should expect to be told what math is. And since just about all of your posts are based on your incorrect interpretations of what math is, it's highly relevant.

[Organic]

master_coda,

Please look here at the complementary association between multiplication and addition of number 4:

http://www.geocities.com/complementarytheory/ASPIRATING.pdf


and please tell me if now the complementary association is understood.


Thank you.

[matt grime]

Firstly ET.

This is approaching a standard I might call legible. Indeed the langauge is far clearer than anything that has preceded it. One wonders how you can use 'hierarchic' and still, depsite me explaining it to you, misuse 'aspirate'.

In ET you start by looking at some partitions of numbers, and how one might combine them modulo some ambiguous handwaving about types. This sort of idea is already known and extensively studied - look at anything on A-infinity algebras.

You are doing well up until the diagrams of swirly circles, and all of a sudden you start using hand-wavy undefined terms such as

'fading transition'

what is a fading transition in rigorous mathematical terms.
Incidentally you seem unsure about rigour. I mean, you call me unimaginative when I demand that you adopt rigorous attitudes, yet insist your pictures are rigorous. Pictures aren't. How do I know you've drawn it correctly? Example: prove that a regular icosohedron exists. Making one out of paper is not sufficient.

Other undefined terms:

structural-quantitative

information point

uncertainty

redundancy


So let's talk aspirating.


1. Given the several representations of a given number as partitions, how do I know which one to chose to multiply together? ASsuming that your multiplication operation is well defined?

2. What is {{},{__}}?

You still havent properly defined this.
And if you think you have, then it will be a s imple task to cut and paste the definition here won't it?

3. If it is the set of all 'numbers' int your system (that can't be the definition can it?) then how does this beat the 'definition' of number as we know it? I mean I can simply say a number is anything in the set N of natural numbers can't I?


4. Aspirating means something else in the English language. I know that is necessarily important, lots of words have dual meanings in mathematics and the rset of the world, but you meant it to impliy aspire. Oh, and you've not defined aspire properly: approaching but cannot become closer to? I don't think that does it:

4a define approach
4b define closeness


5. So, is 4 the set of all of these sub-partitions of 4 things? If so does, say, x*y give you all the sub-partitions of x*y in terms of x and y? Note that you are using the normal definition of 4 in your statements, so you are relying on ordinary mathematics which doesn't define 4 in your opinion to define 4. Thus it is still undefined.

[Organic]

Matt,

'{' and '}' are only framework (a stage) to explore ideas, nothing more, nothing less.

By using {} the idea of emptiness examined.

By using {__} the idea of its oppsite examined.

Any form of {x} is only an x-model of some examined x-idea.

Therefore no x-model is x-itself.

These are my basic rules in this game.

{} and {__} CONTENTS cannot be used by any information system, therefore they are the unreachable limits of any information system.

The logic of the associations between these opposites can be found here, including rigorous models that clearly showing the difference between other logical systems (i added one more model at the and that clearly showing it):

http://www.geocities.com/complementarytheory/BFC.pdf


I construct the natural number by complementary associations between its integral side (its sum) and its differential side (some finite collection of 1’s).


By ZF set theory we know that {a,a,a,b,b,b,c,c,c} = {a,b,c}

It means that concepts like redundancy and uncertainy are not used as fundamental concepts.

The level of where we start using some concept is very very important.

When we allow these concepts to be inherent properties of set's concept, then we enrich our abilities to use set's concept, for example:

<-Redundancy->
c c c ^<----Uncertainty
b b b | b b
a a a | a a c a b c
. . . v . . . . . .
| | | | | | | | |
| | | |___|_ | |___| |
| | | | | | |
|___|___|_ |_______| |_______|
| | |


Where:

c c c
b b b
a a a
. . .
| | |
| | | = {a XOR b XOR c, a XOR b XOR c, a XOR b XOR c}
| | |
|___|___|_
|

b b
a a c
. . .
| | |
|___|_ | = {a XOR b, a XOR b, c}
| |
|_______|
|

a b c
. . .
| | |
|___| | = {a, b, c}
| |
|_______|
|

I think that any iprovment in set's concept has to include redundancy and uncertainty as inherent proprties of set's concept.

The above point of view leading me to what I call Complementary logic, which is a fading transition between Boolean logic (0 Xor 1) and non-boolean logic (0 And 1), for example:

Number 4 is fading transition between multiplication 1*4 and
addition ((((+1)+1)+1)+1) ,and vice versa.

These fading can be represented as:

(1*4) ={1,1,1,1} <------------- Maximum symmetry-degree,
((1*2)+1*2) ={{1,1},1,1} Minimum information's clarity-degree
(((+1)+1)+1*2) ={{{1},1},1,1} (no uniqueness)
((1*2)+(1*2)) ={{1,1},{1,1}}
(((+1)+1)+(1*2)) ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1) ={{1,1,1},1}
(((1*2)+1)+1) ={{{1,1},1},1}
((((+1)+1)+1)+1) ={{{{1},1},1},1} <------ Minimum symmetry-degree,
Maximum information's clarity-degree
(uniqueness)
============>>>

Uncertainty
<-Redundancy->^
3 3 3 3 | 3 3 3 3
2 2 2 2 | 2 2 2 2
1 1 1 1 | 1 1 1 1 1 1 1 1 1 1
{0, 0, 0, 0} V {0, 0, 0, 0} {0, 1, 0, 0} {0, 0, 0, 0}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
| | | | |__|_ | | |__| | | |__|_ |__|_
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|__|__|__|_ |_____|__|_ |_____|__|_ |_____|____
| | | |
(1*4) ((1*2)+1*2) (((+1)+1)+1*2) ((1*2)+(1*2))

4 = 2 2 2
1 1 1 1 1 1 1
{0, 1, 0, 0} {0, 1, 0, 1} {0, 0, 0, 3} {0, 0, 2, 3}
. . . . . . . . . . . . . . . .
| | | | | | | | | | | | | | | |
|__| |__|_ |__| |__| | | | | |__|_ | |
| | | | | | | | | | |
| | | | |__|__|_ | |_____| |
| | | | | | | |
|_____|____ |_____|____ |________| |________|
| | | |
(((+1)+1)+(1*2)) (((+1)+1)+((+1)+1)) ((1*3)+1) (((1*2)+1)+1)

{0, 1, 2, 3}
. . . .
| | | |
|__| | |
| | |
|_____| |
| |
|________|
|
((((+1)+1)+1)+1)

Multiplication can be operated only among objects with structural identity .

Also multiplication is noncommutative, for example:

2*3 = ( (1,1),(1,1),(1,1) ) or ( ((1),1),((1),1),((1),1) )

3*2 = ( (1,1,1),(1,1,1) ) or ( ((1,1),1),((1,1),1) ) or ( (((1),1),1),(((1),1),1) )



Matt, can you give me some address where i can read about A-infinity algebras?

Thank you.



Organic

[matt grime]

Your multiplication is not well defined then, because your 'numbers' are not well defined, that is there is no single element that represents it uniquely. For instance you give me several different answers for 2*3. To deduce anything from this you must acknowledge that '3' is not a number in your system and to call them numbers is misleading. And thus to liken it to ordinary multiplication is also pointless, as is expressing opinions about the so-called lack of distinction between addition and multiplication in ordinary mathematics.

As well as not answering any questions that were asked, you now introduce another undefined concept, 'opposite'. Something you've repeatedly failed to define in the past, relying on intuition. Well, I choose to define it with the empty meaning and conclude your system is vacuous, until such time as you offer a non-intuitive definition.

And, no *I* won't tell you anything about A-infinity algebras. I don't beleve that will accomplish anything as you've yet to understand the basics of common mathematics, and these are very advanced concepts.

[Organic]

Matt,

If you wrote this:

Your multiplication is not well defined then, because your 'numbers' are not well defined, that is there is no single element that represents it uniquely. For instance you give me several different answers for 2*3. To deduce anything from this you must acknowledge that '3' is not a number in your system and to call them numbers is misleading.


Then I have to say that you are much more closed system then I thought, that has 0 ability to understand new fundamental ideas about what numbers are.

You are in your closed system cannot understand how multiplication and addition are complement operations that do not changing the quantity of the number, but only its internal scructure, by breaking and unbreaking its internal self symmetry.

Now, I see you jump and saying: "here comes more undefined terms".

All the rigoruos terms are infront of your eyes but for you terms is written text, not for me.

Also I looked at:

http://arxiv.org/PS_cache/math/pdf/9910/9910179.pdf

http://arxiv.org/PS_cache/math/pdf/0108/0108027.pdf


They do not based on my fundamental ideas, therefore cannot express my mathematical system.

[matt grime]

Let me explain what I as a mathematician would like to see:

you use the labels 2,3 etc and call these numbers. Do we agree on that?

each of these labels corresponds to some set of partitions. Do we agree on that?

when you tell me what 2*3 is, you give me two different answers. So when you ask me to demonstrate the difference between addition and multiplication in boolean logic, I'm entitled to point out that you do not have a well defined multiplication on your 'numbers' because your number 'x' is some set of partitions. So you are asking me to compare chalk and cheese. I think you perhaps don't know what the term 'well defined' means.

this could be interesting combinatorics, but to call them numbers and imply something to do with counting is misleading.
You could demsonstate some interesting combinatorial facts about them.

Here are questions to answer:

1. Given two elements x and y of some 'number' and some other element z of a different number,
does x*z = y*z imply x=y?

2. If you take all x representing some number X and combine them with all y representing some number Y, to you get all elements of XY (odinary multiplication).


In fact, only if 2.is true can you claim to have a proper multiplication, I would suggest.

Your numbers aren't the numbers of mathematics, got it? By definition multiplication of ordinary numbers commutes, you are doing something else. If you stopped trying to call it by the names of ordinary maths you might not meet such hostility.

You are performing some operations on the sets of certain kinds of numbered trees.

you might wish to demonstrate closure, identity, inverse, associativity etc if they exist/are true. then perhaps when you've verified those we can find out what these ought to be called - it perhaps will be some monoid or semigroup.


I@m not closed off to new ideas, as i hope this reply demonstrates. I think you are using the wrong words and looking at the wrong things. if you stepped away from the philosophy then you might get some where when talking to mathematicians who naturally balk at phrases like 'approach but gets no closer to' when presented as you have.

[Organic]

Matt,


Your numbers aren't the numbers of mathematics,


You still don't understand what i am talking about.

1) There is no such objective thing called "Mathematics", it is only a rigorous agreement between some group of people in some period of time, no more no less.

2) Its most basic paradigms can be changed from today to tomorrow , when new fundamental ideas air their view.

3) Let us not forget Godel.

4) The new Natural numbers have the structure of a wavicle, therefore concepts like symmetry, redundancy, uncertainty , entropy and so on, are their inherent properties.

5) Boolean logic is an obsolete logical system for more then 70 years, and logical systems like Complementary Logic are going to take their place in the near future.

6) Theoretical mathematicians like Alain Connes (http://www.alainconnes.org/) say it, and a lot of great physics theoretical mathematicians developing an unordered tailor made pieces of non-Euclidean methods that helping them to solve their problems.

7) Only pure mathematicians are still dreaming in their “Cantor’s paradise” (which is by the way a very ironic name, gave by a community of people that sent Cantor to Mad House for the rest of his life because of their “rigorous” attitudes).

8) Realty and Philosophy were, are and will be the source of any meaningful Mathematical system.

[matt grime]

Originally posted by Organic
Matt,



You still don't understand what i am talking about.

1) There is no such objective thing called "Mathematics", it is only a rigorous agreement between some group of people in some period of time, no more no less.

2) Its most basic paradigms can be changed from today to tomorrow , when new fundamental ideas air their view.

3) Let us not forget Godel.

4) The new Natural numbers have the structure of a wavicle, therefore concepts like symmetry, redundancy, uncertainty , entropy and so on, are their inherent properties.

5) Boolean logic is an obsolete logical system for more then 70 years, and logical systems like Complementary Logic are going to take their place in the near future.

6) Theoretical mathematicians like Alain Connes (http://www.alainconnes.org/) say it, and a lot of great physics theoretical mathematicians developing an unordered tailor made pieces of non-Euclidean methods that helping them to solve their problems.

7) Only pure mathematicians are still dreaming in their “Cantor’s paradise” (which is by the way a very ironic name, gave by a community of people that sent Cantor to Mad House for the rest of his life because of their “rigorous” attitudes).

8) Realty and Philosophy were, are and will be the source of any meaningful Mathematical system.


Numbers: an integer is either the result of adding 'one' repeatedly, or one can produce set descriptions to do with induction and cardinals. They are the units of counting. Demonstrate how to count sheep with your objects.

There maybe some theoretical and philosophical points to answer in the rest of your reply, but they don't seem very relevant to your constructions - everything you mentioned about other people and topics is rigourous and well defined.

Why do you introduce Alain Connes into the discussion? If you wish to talk about non-commutative geometry then I'm very happy to seeing as I've been following different aspects of it for the last 5 years - moduli spaces and C-star algebras at various points.


In fact very few mathematicians go along with Euclidean-geometry as the most useful (the most natural perhaps): as anyone can tell you Hyperbolic geometry is perhaps the most ubiquitous. Actually, that's a misleading statement - do you know the monodromy theorem for Riemann surfaces? There are few complex analytic structures that can be obtained from modding out be symmetries of the riemann sphere, and complex plane, but many more from the Poincare disc.

Of course the thing about that is that we can simply explain where the differences in geometry arise from: the parallel postualte. Your thinking is currently too badly explained. And when people try to point out what you need to do to clarify issues you either refuse, cite something unrelated or accuse them of not being able to understand.

Now, what's a wavicle please, and what's Cantor's paradise?

And why do you trust Alain Connes, even though all his papers are not written using your philosophy, yet dismiss Bernhard Keller's very good papers on A-infinity algebras?

[Organic]

Hi Matt,

1) I am not talking about Non-Euclidean geometry but on Non-Euclidean mathematics, which is not based on Boolean Logic.

2) The fact that you as mathematician do not know what is a wavicle is another sad example of the state of Pure Mathematics of today that do not aware to the meaning of duality of Quantum objects, which are constructed from complementary link between wave(continuous) and particle(discrete). wave+particle=wavicle.

3) Complementary Logic basic principles is built on this kind of associations between opposite concepts, where one of its private cases (the last object of each complementary number) is the standard natural number (that can be used to count sheep).

4) You forgot to write about Godel's incompleteness proves (by the way I am vary close to prove the incompleteness of 3n+1 problem ( http://mathworld.wolfram.com/CollatzProblem.html ).

5) Hilbert in his 1900 lecture closed it by these words:
----------------------------------------------------------------------------
"The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.

For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly.

But, we ask, with the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.

The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfill this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!"
----------------------------------------------------------------------------

Mathematicians took his 23 problems but omitted the heart of its lecturer about the organic unity of Mathematics.

Complementary Logic is an example of an attitude to seek for the Organic Unity of Mathematics.

6) Professional Mathematicians like you does not have the ability to understand what I talking about because their word is based and closed by Boolean logic. If you go beyond Boolean point of view, then and only then you can understand my rigorous models.

7) "Cantor's paradise" is the transfinite universes.

8) Alain Connes in this lecturer http://www.math.ucla.edu/dls/2001/connes.html said that Boolean Logic is going to be replaced by systems that can deal with the complexity of the real world phenomena.

My friend was in this lecturer and hared him saying this.

9) Gauss ( http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html ) was afraid to talk publicly on non-Euclidean geometry because he knew the influence of it on Euclidean standard mathematics word, and he wished to die peacefully without wars against closed-systems' mathematicians.

10) Hippasus of Metapontum (http://scienceworld.wolfram.com/biography/Hippasus.html ) is another sad example of what (maybe) can happened to people how think differently form their community. About Cantor, I already wrote about him.


I'll say it again clearly and loudly:

Any Information system that can be explored, is changeable because concepts like redundancy, uncertainty and entropy are inherent properties of it.

[matt grime]

The strange thing about all this is I've not actually defended Boolean logic at any point, merely pointed out where you've gone wrong (New Diagonal Argument) in misinterpreting infinity, and in your utter failure to define anything coherently.

I'm very aware of wave particle duality - I've never heard the word wavicle before, that's all.
This is looking like some kind of GRE test:

by 'complementarity'

wave is to particle as multiplication is to addition.

This is a fuzzy definition of complement that you could use to associate anything two things you wished. The physical one is that the particle nature of light cannot explain diffraction and that the wave nature of light cannot explain the dual slit experiment. Together these ideas explain its total nature, so in a sense the are complementary. Explain how the complement of addition is multiplication in this light. (No pun intended.)

I'd be interested to hear exactly what Connes said and in what context. But until such time as you define all the words you use, and admit that you made a mistake in New Diagonal argument I'll keep plugging away at it all. Connes is after all a mathematician and likes rigour. I guess your friend doesn't have a transcript of the talk, and in my experience people often hear what they want to hear.

The trees you have defined look like interesting objects, and I would like you to explore their properties (not the philosophy). If I may formalize this for you:

For each n in the Natural numbers, define [n] to be the set of trees you produce.

I don't recall the details, but there exists operations + and * on elements of these sets. Moreover, there exists a distinuished element in each [n] such that they are a model for the natural numbers with * and + mult and add as we understand them, that is there is a map from N to what we'll label [N] respecting the addition and mult of N

Does the composition of two allowable trees produce an allowable tree? Does the comp of all trees in [n] and [m] produce all allowable trees in [nm]or [n+m]? I recall that you don't allow all possible numbered trees, but I could be mistaken.

This seems like a good thing to examine, with genuinely interesting possibilities. It could be s subalgebra of an A-infinty alg, or similar, but you would need to insert some action of symmetric groups in there. Does the distributive property hold.

[n] is your complementary number. Now that you've put the word complementary in there you have removed one of the criticisms.

Is * or + associative?

what are the identities? are there inverses, what if you append formal inverses?


You need to improve your standard of exposition to remove ambiguity, and it would be beneficial for you to stop relying on emotive descriptions and alleged intuition and just say what something is.

[Organic]

Matt,

There is a lot of work to be done on Complementary Logic.

Again, the main idea here is this:

Any Information system that can be explored, is changeable because concepts like redundancy, uncertainty and entropy are inherent properties of it.

Incompleteness is welcome by this point of view.

If you start asking questions, then use your professional skills and examine it by yourself.

[matt grime]

I've got enough research of my own to be doing at the moment, I thought you might benefit from some suggestions. After all the statement that there is a multiplication implies that the object x*y*z is well defined as it implies associativity. I don't recall seeing a proof of that - which isn't to say that you haven't provided one, but that I don't remember seeing it. In particular you need to prove you assertion that the natural numbers are recoverable as the dsitinguished elements {{{{{{1},1}1}...}

or what ever it is, and that means demonstrating closure, associativity mult indentity, etc.

As for the other stuff, you'd need to state what you mean by entropy, uncertainty and redundancy before I could comment. I know what entropy is as a concept, and I know what it is a mathematical quantity in statistical physics etc, but I don't immediately see the relevance here.

[Organic]

It is not associative, for example:


(2*3)*4 =
( ((1,1),(1,1),(1,1)), ((1,1),(1,1),(1,1)), ((1,1),(1,1),(1,1)), ((1,1),(1,1),(1,1)) )

2*(3*4) =
( ((1,1,1),(1,1,1),(1,1,1),(1,1,1)), ((1,1,1),(1,1,1),(1,1,1),(1,1,1)) )

[matt grime]

which representatives of [2] [3] and [4] are those you've chosen?

Is it associative for the choice of 'last in the set' that corresponds to the natural numbers? Or are you claiming a different multiplication on the Naturals that is not associative?

[Organic]

The algorithm to draw what you call [n] is:

( It was written by stratman in http://www.physicsforums.com/showthread.php?s=&threadid=12436 )

For any n > 0, the following algorithm calculates the number of transitional states:

ASSOCIATION_LEVEL(int n):Vector

1 V:= new vector
2 if n = 1 then
3 V:=V + the single root of level 1
4 if n = 2 then
5 V:=V + the two roots of level 2
6 else
7 if n >2 then do
8 For each partition vector P in n do:
9 ROOTS(P,V)
10 return V

ROOTS(vector P,vector V)

1 roots :=[]
2 for each element in P do:
3 if element is a root then
4 roots:= roots + element
5 else
6 roots:=ASSOCIATION_LEVEL(integer value of element)
7 return horizontal alignment of the Cartesian product of P and draw new arc around them

By using Cartesian product we get some left-right structures that must be omitted from each [n] level.

You can use this algorithm here on [1] to [7]:

http://cyborg2000.xpert.com/ctheory/



please look at this paper from its introduction chapter (page 7 in the paper, which is page 8 of the acrobat reader):

http://www.geocities.com/complementarytheory/CATheory.pdf

Please read until the end of CATheory.pdf


About entropy:

Prime numbers are the anti-entropy structures because no first partition-level of them is settled down to equal parts (accept the first structure of the form 1*n).


Thank you.


Organic

[Organic]

Matt,

The main idea of the pervious post is:

Probability is fundamental property of the Number concept.

[matt grime]

Originally posted by Organic
Matt,

The main idea of the pervious post is:

Probability is fundamental property of the Number concept.


That's your NUMBER not mine there is it? And I asked you to define entropy, instead you define anti-entropy in a manner of speaking. That isn't sufficient. Telling me a cat is not a dog doesn't tell me what a cat is.

[Organic]

Matt,

1) From your reply it is easy to show that you did not read my last 2 posts to you.

2) If you read my last post then you can see that Entropy is any partition to equal parts.

[matt grime]

Originally posted by Organic
Matt,

1) From your reply it is easy to show that you did not read my last 2 posts to you.

2) If you read my last post then you can see that Entropy is any partition to equal parts.

I have. And I did again just now, and if you'd like to cut and paste the sentence I'm evidently blind to in which the words 'entropy is..' appear.

Her'e what you do say:

About entropy:

Prime numbers are the anti-entropy structures because no first partition-level of them is settled down to equal parts (accept the first structure of the form 1*n).

That alone does not meet the criterion of defining entropy. It only gives a necessary condition, not sufficient.

It is possibe to deduce what entropic implies, but not what implies entropic. And that is only true assuming that something must be either entropic or anti-entropic, which I don't know to be true either.

[Organic]

Some examples:

Partition's first level of 5 is:

1,1,1,1,1 <----ENTROPY
2,1,1,1
2,2,1
3,1,1
3,2
4,1

Partition's first level of 6 is:

1,1,1,1,1,1 <---- ENTROPY
2,1,1,1,1
2,2,1,1
2,2,2 <----ENTROPY
3,1,1,1
3,2,1
3,3 <----ENTROPY
4,1,1
4,2
5,1

Partition's first level of 7 is:

1,1,1,1,1,1,1 <----ENTROPY
2,1,1,1,1,1
2,2,1,1,1
2,2,2,1
3,1,1,1,1
3,2,1,1
3,2,2
3,3,1
4,1,1,1
4,2,1
4,3
5,1,1
5,2
6,1



From the above examples we can clearly see the difference of the entropy's first partition's levels between composite and prime numbers.

[Organic]

Each number (5,6,7) in the previous post is an association levels of | and _ , which can be found here:
http://www.geocities.com/complementarytheory/P0is1.pdf

[matt grime]

ok, so entropy is the division of a number into equal parts.

and?

i still maintain that you might have some interesting structure on trees. i don't really care too much about them. this at least constitutes a case where you've answered, albeit badly, a question.

you've still not addressed the issue that there is a mistake in your new diagonal argument, nor have you bothered to explain adequately what your theory of numbers is - it relies upon the extant definition, which you think is inadequate for example.

[Organic]

Matt,

If you want to understand my point of view about the diagonal argument, you have no choice but to open your mind to my new point of view about numbers.

Symmetry is the main player in my numbers, and where symmetry is probability cannot be ignored.

Please read this (including its links):

http://www.geocities.com/complementarytheory/Identity.pdf


Thank you.

[matt grime]

No, organic, I don't. You presume to use 'the axiom of infinity induction' to complete some list of 'combinations'. as I have explained to you at length, that is wrong. and presumptious.

incidentally, i think the the person who coined the phrase cantor's paradise, that you think is wrong is the same person that you've taken your name from - david hilbert, who you've cited several times in your defence.

[Organic]

You presume to use 'the axiom of infinity induction' to complete some list of 'combinations'

This line shows that you simply don't understand what I am talking about.

The result of connecting the world 'complete' to infinitely many objects is nothing but an utter bull**** (in your language).

Not my point of view but your mathematical conventions define this utter b*** and give it a nice name like "aleph0" which is the first transfinite object that can exist because there exists a complete list of all N objects.

My proof simply shows that this approach about infinity must lead to a self contradiction result like (2^aleph0 >= aleph0) = {}.

My point of view is simpler and richer than your point of view because symmetry and probability are involved in my point of view and in your point of view they are ignored.

Without symmetry and probability you cannot deal with infinity.

Because you choose to ignore this we have nothing to communicate about.

You stay in your private and cozy world refusing to deal with internal complexity of the number concept, and I’ll move on to maybe find mathematicians that do not afraid to look beyond the current conventions.

[matt grime]

And that demonstrates that you don't understand the maths:

(2^aleph0 >= aleph0) = {}.

makes no sense.

I also apologise for losing my temper.

[matt grime]

"aleph0 which is the first transfinite object that can exist because there exists a complete list of all N objects."

No, Aleph_0 is by defintion the cardinality of the natural numbers. Sets have cardinalities. We offer a defintion here nothing else. By the axiom of choice, any infinite set contains a countably infinite subset, therefore it is not strange to decide that Aleph_0 must be the first infinite cardinal. If you want to take issue with this then it is with the fact that there are larger cardinals.

[matt grime]

Ok, let us stop using the word complete as it is evidently causing problems,

do you admit to using the infinty axiom of induction on the finite combinations list to construct and infinite one? how do you do this? you do not provide any method for this construction. what do you mean by the axiom of infinity construction? you do use that phrase, as anyone can check:

http://www.physicsforums.com/archive/topic/10675-1.html

you conclude rightly that no (infinitely) long (enumerable) list can contain all the possibly strings of 0s and 1s.

BUT

you then assert that every combination must be on the list (no new combination is produced, in your words). this contradicts what you've just demonstrated. the only way for this to be true is if indeed the list omitted no elements, but you'd have to prove that - you've not done so, cannot do so, and have in fact proved the negation of that statement. why do you insist that you produce no new element, when clearly you do!


the whole argument rests on the assumption that the new element produced by the diagonal argument is on the list, somewhere off the bottom, like the finite cases.

this is wrong, and just you misusing the diagonal argument one of the simplest proofs there is in mathematics. perhaps its simplicity is what people can't cope with.

you map from the list of 01s sends a string to the binary expansion. this can only work for strings with finitely many non-zero elements, so it cannot possibly contain all the strings! you've not got round this problem either.

i'm not sure how i can put it any more plainly than that.

oh, and telling mathematicians cantor is wrong is a good way for them to not listen to anything else you say. even your own favoured Hilbert thought it was right.
hell, it is right.

[matt grime]

i'm doing these one point at a time.


if you're going to insist on using probability, then you'd better know that that's an axiomatized system too! in fact for continuous random variables you'll meet countable additivity. bugger, eh?

[Organic]

1) aleph0-1 = 2^aleph0 from transfinite point of view.

2) a >= b by Boolean logic cannot be but {}.

3) ...111111111 cannot be in the collection because when it is included in the collection, the collection is complete, and there is no such a thing like: "A complete collection of infinitely many elements".

4) ...111111111 is not just some object but an information structure which exists upon infinitely many scales, and to say that this object is not in the collection, is exactly as if we say that any collection of infinitely many objects has always infinitely many scales to fill.

[Organic]

Symmetry and probability have to be used right from the most fundamental level of any mathematical system, and I mean from the level of the set concept, or form the natural numbers.

[matt grime]

Originally posted by Organic
1) aleph0-1 = 2^aleph0 from transfinite point of view.

No.

2) a >= b by Boolean logic cannot be but {}.

No. one is a set the other isn't.

3) ...111111111 cannot be in the collection because when it is included in the collection,
the collection is complete, and there is no such a thing like: "A complete collection of infinitely many elements".

? There are other strings with infinitely many ones in them than that.

4) ...111111111 is not just some number but an information structure which exists upon infinitely many scales, and to say that this object is not in the collection, is exactly as if we say that any collection of infinitely many objects has always infinitely many scales to fill.


..1111 is not even a number, never mind 'just a number'

I'd not use 'we' when you mean 'just you'

[matt grime]

Originally posted by Organic
Symmetry and probability have to be used right from the most fundamental level of any mathematical system, and I mean from the level of the set concept, or form the natural numbers.

Probabiltity requires the assignment to an event of a number between 0 and 1. It therefore belongs to a 'higher' state of thinking than the naturals because to define it rigorously requires the existence of the naturals, then the rationals, then the reals, as well as a set theory already, as a probability measure is defined on subsets of a proability space. It is not prior to sets as it requires sets for its definition. OF course if you're offering some hand waving 'intuitive' notion of probability then you're on your own again.

[Organic]

Matt,

I am talking about a mutation in the natural numbers, that changes the Math paradigm about them.

[matt grime]

so you weren't talking abour probability then? were you lying, obfuscating, confused or mistaken? Would it be overly scholastic of me to demand you explain why you've chosen the word mutation, adding yet another ill-defined term to the mix? Apart from your evident desire to confuse the issue so that you never explain anything that's asked of you.

[Organic]

You play with the words I use instead of try to understand their meaning.

This is what I call scholastic approach.


By mutation I mean that there is a change in some already used thing that deeply change its proprties.

[matt grime]

Almost never do the words you use have a well-defined mathematical meaning, and when they do it is never in the sense that you are using them.

So, I know what mutation means, now you need to explain what mutation you are talking about. Try and be self-consistent: thus far that has eluded you becaue you keep introducing more concepts that you can't explain before sorting out the previous problems.

Before stating that current maths is wrong, perhaps you should understand what it says? I mean you've demonstrated in the last few posts that you don't know what 'equals' means, nor what probability is, that you don't understand the definition of aleph-0, that you don't know what the valid operations on cardinals are, that you believe there is something called the axiom of infinity induction that we all know and love, and that you can't understand the idea of the proof of cantor's diagonal argument. if you can'it understand these, and because of this lack of understanding decide to write a new theory without defintions, then what are we supposed to do?

[Organic]

No Matt,

I started to write my ideas after I understood that pure mathematics have no ability to deal with complex systems.

And the reason is very simple, it does not distinguish between x-model and x.

I say: x=model(x)

And by this approach my theory do not run after its own tail, creating virtual paradoxes like Russell's paradox, forcing words like "complete" and "all" on infinitely many elements and by this, creating and using almost all its energy to explore the results of this forcing, which is the transfinite words.

When you don't understand my results when using the built-in induction of the axiom of infinity on a sequence of 01 notations, you clearly show that you don't have the most important things that a good mathematician needs, which are:
sense of symmetry and sense of simplicity that implies sense of beuty.

Technical abilities are only tools but because a lot of mathematicians like you do not understand the difference between x-model and x, they become closed systems under their own technical conventions and cannot see anything beyond it.

You are a good example of such a mathematician.

Term: ( http://mathworld.wolfram.com/Term.html )
In logic, a term is a variable, constant, or the result of acting on variables and constants by function symbols.

It is nice isn't it?

But standard math does not use the difference between variable and constant when it define the set concept, and the result is:

By the axiom of the empty set we clearly see that x cannot be but a non-empty object(=something), and only then we can define {}.

By not putting any symbol between '{' and '}' we look at emptiness as a constant.

If x is not empty then x symbol exists.

If x is empty then x symbol does not exist.

And you call this approach an abstract Math?

[matt grime]

I notice that you do not answer the charge that you misuse notations.

First, the whole point of ZF is that it avoids Russell's paradox.

Simplicity and symmetry have nothing to do with you misusing the axiom of infinty (which merely states that an infinte, indcutive set exists, namely N).

The rest of your post is rambling and incoherent, in particular the assertion that the emtpy set is what you get when you put nothing in between some parentheses.

[matt grime]

OH, and given that you apparently cannot even understand the simplest statements about mathematics (Collatz 3n+1 thread), I find you casting aspersions on my abilities just a little offensive. So, explain the difference between the empty set and the model of the empty set.

And, btw, I'd be the first to say I'm not a good mathematician.

[Hurkyl]

By not putting any symbol between '{' and '}' we look at emptiness as a constant.

Again, it sounds like you're talking about lexical analysis, not set theory. The null string is indeed a constant in that context. And, of course, if &xi; is a variable representing a string, then &xi; may certainly be the null string, and if we have a proposition that looks like \forall \xi: P(\xi) (where &xi; ranges over strings), then P(\epsilon) must be true (where I've chosen to use the symbol &epsilon; to denote the null string).

But do beware; when &xi; is a variable representing a string, the string '{&xi;}' does not denote the set {&xi;}. (The former is the string denoted by &xi; surrounded by braces, and the latter is the set containing the string &xi;)

For a concrete example, the string

'{a, b}' is not the set {'a, b'}. (where a and b are not string variables) The former is a string that denotes the two element set {a, b}, while the latter is a set containing the string 'a, b'.

[Organic]

Hi Hurkyl,

I am not talking about strings or representations.

I am talking about the difference between theory of x and x.

x=x is a tautology, and this point of view does not distingiush between x and model(x).

Mathematics is a theory, therefore any x=model(x).

By this approach the right framework is {model(x)} so no actual x is involved.

When any x is model(x) no x property has an impact on the existence of the framework itself.

for example:

x=model(nothing)

x=model(something)

or if you like:

x=theory(nothing)

x=theory(something)


In both cases x is a theory of x, and we can avoid the paradoxes that caused by x=x point of view.

[matt grime]

Originally posted by Organic
You play with the words I use instead of try to understand their meaning.

This is what I call scholastic approach.


By mutation I mean that there is a change in some already used thing that deeply change its proprties.

Teach by example:

let S by the subset of Natural numbers that are uninteresting. S has a least element (not all numbers are interesting), yet this least element is now interesting as it is the least interesting positive number.

There are many such paradoxes lying around all of the to do with imprecise definitions.

But not only do you use imprecise definitions but fail to use correctly the precise ones that exist.

Clear up the mistakes in New Diagonal before committing any more

[Organic]

You are talking to yourself, not to me because of a very simple reason.

For me the meaning of "what is a number?" starts here:

http://www.geocities.com/complementarytheory/count.pdf

For you numbers are objects that do not depends on your ability
to define them.

For you mathematical objects are actual objects, based on unchanged
logic terms.

For me Mathematics is only a form of communication (a language) that totally depends on our abilities to develop it.

From this point of view Mathematics is an open system that can deeply be changed when its paradigm is changed, and paradigm is not an actual object, but only a theory.

Please read again my answer to Hurkyl:
---------------------------------------------------------------------------
I am not talking about strings or representations.

I am talking about the difference between theory of x and x.

x=x is a tautology, and this point of view does not distinguish between x and model(x).

Mathematics is a theory, therefore any x=model(x).

By this approach the right framework is {model(x)} so no actual x is involved.

When any x is model(x) no x property has an impact on the existence of the framework itself.

for example:

x=model(nothing)

x=model(something)

or if you like:

x=theory(nothing)

x=theory(something)


In both cases x is a theory of x, and we can avoid the paradoxes that caused by x=x point of view.
---------------------------------------------------------------------------

The ZF axiom of the empty set does not see mathematics as theory, but
look at its conclusions as actual results, for example:

If x is not empty then x symbol exists.

If x is empty then x symbol does not exist.

This is a primitive and definitely not an abstract approach.

The same is for the transfinite system, that forcing the theory of infinity to some actual object that its cardinality can be captured by forcing "for all" on the theory(=model) of infinity.

When forcing "for all" on the theory of infinity, you have no theory but actual infinity like "emptiness itself" or "fullness itself".

Both states are beyond any theory, therefore they are the limits of any theory.

Shortly speaking, no information can be exchanged between the actual infinity and the theoretical infinity.

The form of theoretical infinity cannot be but "infinitely many elements".

No infinitely many elements can be an actual infinity.

Your precise Math do not understand this, therefore it is using "for all" on theoretical infinity and the result is like driving by using "full gas in neutral".

[matt grime]

Oops, committed some more there haven't you? Like confusing infinite and infinity again.

[Organic]

Infinity: http://mathworld.wolfram.com/Infinity.html

Infinite: http://mathworld.wolfram.com/Infinite.html

In both cases standard Math using the word "Quantity".

Emptiness(=no information) or Fullness(=total information)
have no meaning through quantitative point of view.


Also in your subset of Natural numbers that are uninteresting
there is no paradox, because S = {}.


( By the way can you answer to http://www.physicsforums.com/showthread.php?s=&threadid=13461&perpage=12&pagenumber=2
in 3n+1 problem? )

[matt grime]

Organic, in simple English, not maths, infinite is an adjective, and infinity is a noun. You mix up the two at will.

As for the paradox, it is only in YOUR opinion that there are no 'uninteresting' numbers. I didn't define what interesting meant so how can you possibly tell me what is or isn't interesting?

[matt grime]

As for the request to reply to the other topic - no I don't see the need to. If you'd read the damn link you gave you'd answer your own question.

[matt grime]

New technique:

Organic, you use the axiom of infinity induction, what is this? If you are claiming to use the axiom of an infinite set, then explain why, because it just states that an inductive set exists.

Until you define it, or justify it, I will post this question to every reply.

[matt grime]

Damn, this is too easy sometimes. But you've either contradicted yourself OR you don't know what a tautology is OR you don't know what = means. You shouldn't have nicked that phrase off me if you didn't know what it meant. You see in the current thread in general math you insist that x=x is only true under certain conditions - either x is actual or theoretical, and not the other. Here you say it is always true. Damn, that was a silly mistake wasn't it?

[Hurkyl]

For me Mathematics is only a form of communication (a language) that totally depends on our abilities to develop it.

One would surmise that such a belief would lead to:

(1) a desire to understand what has already been developed.
(2) a desire to learn how to communicate comprehensibly.

Instead, you

(1) make up your own meanings for what has already been developed.
(2) that it is everyone else's problem that they can't comprehend you.

[Hurkyl]

You say things like

By not putting any symbol between '{' and '}' we look at emptiness as a constant.

And

I am not talking about strings or representations.

Based on this, I assume you are not referring to ZF at all in this thread. Is this correct?

[matt grime]

As often is the case, it is rather unclear what is being talked about, as questions provoke responses that demand more questions.


Here is as synopsis of what we've established.


The axoims of ZF are 'incorrect', in organic's opinion because of his view that the axoim for the empty set contains something unacceptable.

In response one might expect another thoery to be put forward. But there isn't one.

The point arises in a semantic, non-mathematical, argument about emptiness, whatever we are supposed to assume that means.

The result is a series of bizarre postings containing little in the way opf recognizable mathematics.

What is clear is that Organic feels the logic of current mathematical thinking is inconsistent. What is also clear is that he doesn't understand much maths, as evinced by his reasoning that 'boolean logic can't cope with infinity', and the following deductions about the requirements of probability, yet he hasn't offered a way to define probability without relying on things he finds inadequate.

So, organic has posted something he states to be a theory for sets, though it seems incomplete. He also doesn't clearly understand what a model is, thinking that the sets we use somehow 'are' ZF, rather than understanding that ZF is a series of rules that our sets obey. There are other set theories out there, each has their own advantages and disadvantages.


For instance, depending on the set theory one uses, one can make different deductions about what the vanishing of ext groups means.

It is quite hard to make sense of it some times, and even harder to make Organic understand what the objections are, especially as I know very little set/model theory.

DOn't know about you, but I feel the goal-posts are constantly shifting in Organic's intents.

[Organic]

The logic of actual and potential infinity:

http://www.geocities.com/complementarytheory/CompLogic.pdf

http://www.geocities.com/complementarytheory/4BPM.pdf



Emptiness:

E=emptiness

oo...-nor-E-nor-E-nor-E-nor-...oo


Fullness:

F=Fullness

oo...-and-F-and-F-and-F-and-...oo



To use these two logical chains as input, we must break them
but when we break them, we no longer deal with actual infinity
but only with potential infinity of "infinitely many objects".

For example: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Potential infinity and actual infinity are preventing each other.

Actual infinity cannot be explored by any theoretical method, therefore it is the limit of any theoretical method.

Potential infinity = theory of(actual infinity), no more.

Also fullness and emptiness are preventing each other, but they also define each other as we can see here:

http://www.geocities.com/complementarytheory/BFC.pdf

By BFC.pdf we can clearly see that Boolean and Fuzzy Logics are private cases of Complementary Logic.

Complementary Logic is based on the symmetry concept and researches its braking levels as natural part of its method.

Therefore numbers are first of all forms of symmetries that are ordered by their internal symmetrical degrees, as we can see hare:

http://www.geocities.com/complementarytheory/ET.pdf

and here:

http://www.geocities.com/complementarytheory/P0is1.pdf


Shortly speaking, the Natural numbers by Complementary Logic are Quantum structures, where the standard Natural numbers are only a one private case of some Quantum structure.

[Organic]

Hi Hurkyl,

Based on this, I assume you are not referring to ZF at all in this thread. Is this correct?

I take ZF as an example of non fundemental thinking about the set's concept, and I show it by ZF axiom of the empty set (x can be nothing XOR something).

[Organic]

Matt,


As for the paradox, it is only in YOUR opinion that there are no 'uninteresting' numbers. I didn't define what interesting meant so how can you possibly tell me what is or isn't interesting?

Yes, you wrote: "The least uninteresting number is intersting"

Therefore S = {}.

[matt grime]

'and' and 'nor' are logical operations defined on conditions that are true or false. So E must be a statement that is true of false, it is not a set. I mean the set of natural numbers, in both our worlds, exists, what does it mean for the set of natural numbers to be true (or false) as a statment in logic? I really should have pulled you up about misusing predicates and quantifiers before.


Standard unanswered question: explain what you mean by using the axiom of infinity induction on the list of combinations in new diagonal argument?

[matt grime]

Originally posted by Organic
Matt,


Yes, you told: "The least uninteresting number is intersting"

Therefore S = {}.

perhaps i should have more clearly written

the smallest 'uninteresting number' is interesting.

what's your point? How can you conclude that the set is empty? i said the set wasn't empty, i said there were uninteresting numbers.

it's a matter of opinion, that's the problem organic. i mean it might be that just being the smallest uninteresting number isn't interesting, but it might be, who can say what counts?!?!

[Organic]

Matt,

Please read this:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

Take out the relevant part that connected to your question and ask a detailed question about it.

Thank you.

[Organic]

Matt,

i said the set wasn't empty

So what.

If the smallest 'uninteresting number' is interesting, then S has no objects in it.

[matt grime]

Originally posted by Organic
Matt,

So what.

If the smallest 'uninteresting number' is interesting, then S has no objects in it.

well, done, you're beginning to see the paradox! the set of uninteresting numbers is empty and not empty. do you get it yet?

[matt grime]

Originally posted by Organic
Matt,

Please read this:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

Take out the relevant part that connected to your question and ask a detailed question about it.

Thank you.


Ok, it's still the same question:

how does one get an infintely long list using 'the axiom of infinity induction'?
that is not a meanignful statement as far as i can tell. this is page 3 paragraph 2. you've never explained this step despite me asking on at least 6 occasions that i can recall off hand.

[Organic]

No Matt,

S = {} or S does not exist by your definition.

If it is not empty then by your definition no object is uninteresting.

Therefore S (by your definition) does not exist, because even if one member of it is interesting then we can find the relation of each member in S to this interesting member, so all members are interesting or S has no members.

Shortly speaking no set can include its negation and survive.

[Organic]

Do you mean to this part?
--------------------------------------------------------------------------
We can get a combinations list of infinitely many places, by using the ZF Axiom of infinity induction, on the left side of our combinations list, by using the induction on the power_value of each column, for example:

2^0, 2^1, 2^2, 2^3, ...
--------------------------------------------------------------------------

[matt grime]

yes i mean that. the axiom of infinity merely tells us that there is something akin to the natural numbers in our model. how does that allow you to produce these strings, and what does it tell you about the strings so produced, and how, for that matter, are you inducting?

[Organic]

ZF axiom of infinty simply says: If n exists then n+1 exists.


Do you agree with that?

[matt grime]

and what's that got to do with what you're doing?

incidentally, in immediate prelude to that there is a mistake (well, there are many...}

you say the right column is 'based on 2^0' which is questionable, because it is 01010101. check, i think you'll find it's 11001010

[Organic]

Matt,

Now I see that you simply don't understand my method.


2^0=1, and 1 tells us after how many times to switch from 0 to 1 or from 1 to 0, therefore the result is 0101010101...

2^1=2, therefore the result is 001100110011...

2^2=4, therefore the result is 0000111100001111...

and so on.

[matt grime]

firstly, i think you'll find that the first column in your paper IS NOT 0101010....

now you've explained that, let us examine what this means - given any row it is immediate that after some point, reading right to left, that every entry is zero. This means that the list you construct has only strings that have finitely many non-zero elements in the. Thus it is some sublist of ALL strings. Clearly it is countable.

If you'd explained your method clearly this would be unimportant.

What IS important is you assertion that the list produced contains every element, that is that the diagonal argument produces something that is NOT added to the list because it is already there. Clearly that is not true. Even if only because befoer that you've misused the diagonal argument.

Incidentally, in what way have you USED the axiom of infinity INDUCTION to do anything?

[Organic]

Matt.

01010101... 001100110011... 0000111100001111... are columns not rows.

Now I clearly see that you simply don't read what I write in my papers.

[matt grime]

look at page 2, the array for n=3 and all the combinations there. you claim that the right hand column is based on 2^1 or something, that is that it ought to be 01010101. it isn't it is 1100101 how can we see that is what you say it is when it clearly isn't?

i know they're columns, where first is the rightmost. look at the rightmost column on page 2 for n=3.

[matt grime]

so let us assume that you've picked some ordering of the rows where you've got this pattern - which you CAN do for n=3, as you've got on page 1 and page 3, but not page 2.

So, we can produce a doubly infinite array (to the left and down) where each column is periodic ok, getting there. this is not using the infinity axiom of induction. whatever that might be - you'vr not explained that either, btw, axiom of infinity yes, but not the axiom of infinity induction.

SO? Reading across in each column gives us an element of the combination list, doesn't it? Well, as I stated before, that tells us that each element on the list, that is each row, has only finitely many non-zero entries.

How are you concluding that this list has all the elements you want on it? Cantor's diagonal argument produces something not on that list, it has infinitely many non-zero entries, you claim that this need not be added to the list because it is already on it. No it isn't - this is the basis for you deciding 2^aleph-0=aleph-0

[Organic]

Again you show us that you don't fully read what I write, therefore don't understand what you see.

Go back to page 1 read all of it and then read cerfully the first lines in the top of page 2.

[matt grime]

Have done, realized what was going on, so see the last post which i reprint here:
so let us assume that you've picked some ordering of the rows where you've got this pattern - which you CAN do for n=3, as you've got on page 1 and page 3, but not page 2.

So, we can produce a doubly infinite array (to the left and down) where each column is periodic ok, getting there. this is not using the infinity axiom of induction. whatever that might be - you'vr not explained that either, btw, axiom of infinity yes, but not the axiom of infinity induction.

SO? Reading across in each column gives us an element of the combination list, doesn't it? Well, as I stated before, that tells us that each element on the list, that is each row, has only finitely many non-zero entries.

How are you concluding that this list has all the elements you want on it? Cantor's diagonal argument produces something not on that list, it has infinitely many non-zero entries, you claim that this need not be added to the list because it is already on it. No it isn't - this is the basis for you deciding 2^aleph-0=aleph-0

[Organic]

Be aware that we have an ordered collection of infinitely many 01 sequences by using the ZF axiom of infinity built-in induction on the power level of 2^power_level.

Again ZF axiom of infinity: If n then n+1.

Therefore the power_level = aleph0 and we have an ordered collection of 2^aleph0 elements.

[matt grime]

you see, now this is where you go horribly wrong, you pass from a finite cardinal to an infinite one. YOU CANNOT DO THIS! It is not valid.

AND, you've still not said what you mean by

the ZF axiom of infinity in-built induction. What the hell is this?!?! On the power level 2^powerlevel, what the hell is power level?!?!

And the axiom of infinity does not state if n then n+1

[Organic]

Please write the ZF axiom of infinity in English words.

[matt grime]

The simple interpretation is that there is an inductive set, that is we can produce a set that behaves like the natural numbers. It is bad english to say that n impies n+1. The maths states that given a set which we can label by n we can form another set which we can label n+1, by taking the union of the set labelled n with the set containing the empty set. It is your english that is most at fault, and virtue of that then your maths. It is the existence of the set indexed by n, with the set containing the empty set that implies there is a set labelled n+1, the labels being the cardinalities.


All you are doing is saying that there infinitely many natural numbers, OK.

let the r'th column of a doubly infinite array be the string of 2^r 0s, then 2^r 1s then 2^r 0s and so on. There is no need to use any axiom of infinity induction (which is...?0 merely that the naturals are infinite! There is no induction! we are not using the s'th level to define the s+1'st level, which is what an induction would be!

[matt grime]

Here is a more mathematical definition.


There is a set W that contains the empty set and if any set y is in W then the set containing the union of y and the set containing y is also in . By induction contains every finite integer.


from:

http://www.mtnmath.com/book/node53.html

[Organic]

My language is Hebrew.

Now Zf axiom of infinity is:

There is a set Omega that contains the empty set and if any set y is in Omega then the set containing the union of y and the set containing y, is also in Omega.

By induction Omega contains every integer.

So as you see, we are talking about Omega = {1,2,3,...}
and Omega is aleph0.

Therefore our collection is 2^aleph0 collection.

[matt grime]

No, aleph-0 is not a set. It is the cardinality of a set. Your omega is the set of natural numbers, its cardinality is Aleph-0.

The last line:

Therefore our collection is 2^aleph0 collection.


Makes little sense to me.

[Organic]

http://us.metamath.org/mpegif/aleph0.html

My mistake. Omega = |{1,2,3,...}|


Therefore we have an ordered collection of 2^aleph0 members.

[matt grime]

No, you do not have a collection of 2^aleph-0 elements, assuming you mean the set of 'combinations' you produce. I do not know how you are concluding this but it isn't 2^aleph_0. This is the cardinality of the power set of of the natural numbers. The list you produce is not in bijection with any set of cardinality 2^aleph-0, it is in fact obviously of cardinality aleph-0.

[Organic]

No Matt,

By using the induction of ZF axiom of infinity on 2^power_value (by mistake i wrote power_level) power_value = aleph0.

[matt grime]

edit: word power inserted at crucial point

OK, the cardinality of the power set of n-elements is 2^n, the cardinality of a POWER set of card alpeh-0 is 2^aleph-0. The list you produce has card aleph-0. The only way you could claim it were 2^aleph-0 was if it were the list of all subsets of N, but it isn't it only contains the finite subsets. This is the crux of the issue - you do not produce a 'list' of 2^aleph-0 elements, you prove it is countable yourself.

The only way it you could do otherwise would be to demonstrate the list you produce by some 'induction' that isn't an induction, contains all 'combinations'; it doesn't!

[Organic]

Again no Matt,

My collection is all possible 01 combinations of 2^aleph0 ordered elements finite an non-finite.

Please read again page 1 in:
http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

and see for yourself that i am talking about P(aleph0).

[matt grime]

But it clearly isn't!

The thing you construct is the doubly infinite array, the first column (right to left) is 0101010...

the second is 0011001100....

the r'th column we now agree is 0000....00111..111000..00... with 2^r 0s and 1s in each segment.

SIMPLE proof this only gives the finite subsets - let x be some combination on the list, say it is the n'th. Since every column after the n'th starts with 2^n zeroes, and 2^n>n for n >1, then clearly every entry after the n'th, reading right to left, is zero, and thus there are only finintely many elements in that subset. THe set was arbitrary.


As you send a string to the binary expansion, to show it's countable, you can only have a finite number of 1s in the damn string!

[Organic]

No Matt,

It gives P(aleph0).

Again you don't read what I wrote in page 1 :

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

Your "PROOF" is not about 2^aleph0 members, because n not= aleph0.

[matt grime]

But your bizarre bastardization for the finite sets doesn't apply in the infinte list.

In fact as you go top left to bottom right in your diagonal argument, can I ask how you use this in the infinite case - which element in the first string (row) do you use?

I can't say it any more clearly, the list you produce, by your own admission only contains strings with a finite number of non-zero entries on it. This is 'your' proof they are countable (they are trivially countable by construction, but you don't seem to realize this).

By your own admission, the element ..1111 is not on the list, yet it ought to be if the list enumerated the power set. And yet you claim it contains all the combinations... bizarre and self contradictory.

[Organic]

Left-right or right-left diagonals holds only for finite P(n).

When I deal with P(aleph0) ordered list then you can see that it is
a right-left diagonal.

It is no problem to say that ...111111 is also in the list but then
we clearly deal with a finite ordered list, which is not our case.

Again, be aware to the fact that we are dealing with an ordered collection.

Also ...111111 is not just a one member but an open interval of aleph0 scales (of 2^aleph0 ordered collection).

[matt grime]

Originally posted by Organic
Left-right or right-left diagonals holds only for finite P(n).

When I deal with P(aleph0) ordered list then you can see that it is
a right-left diagonal.

It is no problem to say that ...111111 is also in the list but then
we clearly deal with a finite ordered list, which is not our case.

Again, be aware to the fact that we are dealing with an ordered collection.

Also ...111111 is not just a one member but an open interval of aleph0 scales (of 2^aleph0 ordered collection).


So, we should looka t the diagonal from top right to the 'bottom left'.


the scales thing is not important - ...1111 corresponds to the element in the power set that is the set N.


That doesn't answer anything important anyway.



The thing you construct is a doubly infinite array from right to left and top to bottom, it contains only strings with a finite number of non-zero elements as I've proved independently of you and as you prove yourself by writing an explicit bijection with 2-adic expansions.

Clearly the list is countable (nb, for mathematicians, lists are countable by definition), yet you insist that it contains all combinations, despite proving it doesn't yourself and repeatedly saying the string ..1111 isn't on it! Nor is ..01010101, nor is ..001100110011 etc.

You make two accurate assertions - that there is no bijection between N and P(N) and that the Finite Power set is countable. The problem is you then say they are the same thing! They are not. You prove this yourself.



And I don't understand why you seem think that N is not an element of P(N) {N is the set of natural numbers}, that is the only way I can read your statement about when ...1111 is a combination.

[Organic]

1) I proved that |P(N)|>=|N| iff |N| is the cardinal of ALL N members.

2) I clime that there is no such a thing the cardinal of infinitely many elements, because they cannot be completed.

3) In this case all we have is (...111,...000] XOR [...111,...000)

4) There is no shuch a thing [...111,...000]

5) I am going to sleep, so see you and have a good night.

[matt grime]

Originally posted by Organic
1) I proved that |P(N)|>=|N| iff |N| is the cardinal of ALL N members.

2) I clime that there is no such a thing the cardinal of infinitely many elements, because they cannot be completed.



Well, you didn't prove 1 assuming N denotes the set of natural numbers, although you assert it with a wrong proof, as you take a fininte cardinal result and put in aleph-0 and claim the answer. Well, here's a counter example:

for all finite cardinals n>n-1 hence aleph-0<aleph-0

and 2. is a defintion! which incidentally you use in part 1. Besides, it doesn't matter what you believe, it matters what you can prove, or disprove. For instance, in what way is N, the set of natural numbers not complete? I mean, there meanings where it is not complete, algebraically. If I list the numbers in N in sequence, which one do I omit? Either give me an example, ro proive I must omit one.

[Organic]

Matt,

PROOF:

Let T be the 01 collection with cardinality of P(N).

Let L be the 01 collection with cardinality of N.

T is:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...

In this private case we can see that because T 01 collection is longer then L 01 collection, ...010101 is not in L but it is in T.

We can change the order of T 01 collection and then we shall find another 01 sequence which is not in L but is in T.

In this stage we can clearly say that |T|>|L|.

The width of T 01 collection = |N|

The lengh of T 01 collection = |P(N)|

Because any missing 01 sequence is already in T, we MUST NOT add it to L.

Therefore we can find a 1-1 and onto between 1,2,3,... to any T member.

But because |L|=|N| we can conclude that |L|=|T|.

Now we have |T|>=|L| which is a contradiction.

Therefore transfinite cardinality does not exist.

Q.E.D


This proof holds for finite or infinitely many objects.

Let us see it in a finite 01 collection.


T has the cardinality of P(3)

L has the cardinality of 3


T 01 collection is:

000
111
001
110
010
101
011
100

01 sequence of 101 is already in T therfore we MUST NOT add it to T.

Therefore there is 1-1 and onto between any 1,2,3,... to any T member:

000 <--> 1
111 <--> 2
001 <--> 3
110 <--> 4
010 <--> 5
101 <--> 6
011 <--> 7
100 <--> 8

[Hurkyl]

Let T be the 01 collection with cardinality of P(N).

Let L be the 01 collection with cardinality of N.

T is:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...

How do you justify writing T as a list?

[Organic]

Hi Hurkyl,

Please read this:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

As you can see I can combine between:

...000000
...000001
...000010
...000011
...
...
(...111,...000]

and

...111111
...111110
...111101
...111100
...
...
[...111,...000)

and get T collection:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...

[matt grime]

I must echo Hurkly here - you are, by writing it as a list and applying the diagonal argument to it presuming it is enumerable AND contains all the 'combinations'; you prove this is not possible. END OF PROOF. But no, you carry on...


More abuses of maths are:

1. The set with cardinality |P(N)|? Don't you just mean P(N)? There are lots of sets with cardinality |P(N)|, you can't talk of THE set...

2. Same as above but with N not P(N) in there

3. you don't know what's in L because you've not said what L is


What is the bijection you claim from T to N? Please say it's not by the numbering of the row.


The finite case does not imply the infinte case. You are attempting to say that becuase, in the finite case we only look at the top nxn square in an nx2^n array, we can do the same in the infinte case - but the construction doesn'tdo this. THere are as many rows as columns in the infinite case, and no row is not used in the diagonal argument.

[Hurkyl]

Let T be the 01 collection with cardinality of P(N).

As you can see I can combine between:

...000000
...000001
...000010
...000011
...
...
(...111,...000]

and

...111111
...111110
...111101
...111100
...
...
[...111,...000)

and get T collection:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...

Why do you think this list has cardinality P(N)?

[matt grime]

Originally posted by Organic
Hi Hurkyl,

Please read this:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

As you can see I can combine between:

...000000
...000001
...000010
...000011
...
...
(...111,...000]

and

...111111
...111110
...111101
...111100
...
...
[...111,...000)

and get T collection:

...000000
...111111
...000001
...111110
...000010
...111101
...000011
...111100
...
...


The first collection gives you those with finitely many 1s in, the second gives you the collection with finitelym any 0s. THey both have card aleph-0 so does their union. But it isn't all of the things in T, only the finite and cofinite ones. At no point is the repetitive string....010101010101 in there.

[Organic]

Because T is a collection of 2^Omega unique memebrs, where Omega
is the result of using the ZF axiom of infinity bulit-in induction on the power_value
of 2^0, 2^1, 2^3, ...

[Hurkyl]

Because T is a collection of 2^Omega unique memebrs

Why do you think that? What is omega? Is it cardinality of the natural numbers in ZF?

Omega
is the result of using the ZF axiom of infinity bulit-in induction on the power_value
of 2^0, 2^1, 2^3, ...

This sentence makes no sense.

I assume you mean "Omega is the result of applying induction to the sequence 2^0, 2^1, 2^3, ..."

But this makes no sense because this is not even close to the form of something to which one applies induction... induction is, essentially, when you compute values of something based on previous values (when the domain of the function is well-ordered)

Here, each value is given explicitly; 2^0, 2^1, 2^3, et cetera. There is no computation based on previous values.

[Organic]

Matt,

Yes ...01010101 is in T but not in L.

[Organic]

Hurkyl,

Omega: http://www.mtnmath.com/book/node53.html

Please look at the axiom of infinity.

[matt grime]

Originally posted by Organic
Because T is a collection of 2^Omega unique memebrs, where Omega
is the result of using the ZF axiom of infinity bulit-in induction on the power_value
of 2^0, 2^1, 2^3, ...

As we established last night, you've not defined 'the axiom of infinity built-in induction'. in fact a quick google for the phrase reveals you're the only person on the entire web as google sees it that has used that phrase. There is the axiom of infinity which merely asserts our set theory has some set that contains the Natural numbers (we do: the natural numbers). There is also the principle of induction, which you don't use.

[matt grime]

Originally posted by Organic
Matt,

Yes ...01010101 is in T but not in L.

But you just told us how to construct T, and the ...010101 is not there. You said to add together the finite and cofinite lists to produce the T collection.


And you've still not adequately define L (or T really), but I think we know what they are. Want to make sure?

[Organic]

No Matt,

The pruduct of using ZF axiom on infinity on the power_value of
2^power_value is an ordered collection of 2^aleph0 unique members.

[Organic]

Matt,

I did not add anything to T I just changed the order of its members
to show you that your "proof" by left endless zeros does not hold.

[matt grime]

Originally posted by Organic
No Matt,

The pruduct of using ZF axiom on infinity on the power_value of
2^power_value is an ordered collection of 2^aleph0 unique members.

Erm, that's not what the axiom of infinity states. I see you've dropped the word induction.

The axiom assures us that there is a set conatining the natural numbers. It does not allow us to whack in an alaph-0 at will. Aleph-0isn't even part of ZF's axioms.


So, explain what you are doing to
2^0, 2^1, 2^2....

the nearest I can get is that you are taking unions of sets of size 1,2,4,8,...

the union clearly is countable.

[Hurkyl]

And anyways, induction doesn't require the axiom of infinity, or even much set theory. Here are some definitions:

Definition: < is a well ordering on a set S iff subsets of S have smallest elements (with respect to <)

Notation: We often abbreviate this by saying "S is a well ordered set" and leave the ordering, <, implicit.


Theorem (induction): If

(1) S is a well ordered set with smallest element s
(2) P(x) is a logical proposition
(3) P(s)
(4) (y in S, y < x, P(y)) &rarr; P(x)

Then

z in S &rarr; P(z)


This barely requires any set theory, and doesn't even require things like "for all" or "there exists"

[matt grime]

Originally posted by Organic
Matt,

I did not add anything to T I just changed the order of its members
to show you that your "proof" by left endless zeros does not hold.


from your original definition in the first new diagonal argument my proof is perfectly valid.


I can now prove that any string has either no 1's after some point, OR it has no 0's given your new way of constructing T by interlacing two clearly countable sets, since it came from one of two original lists where one or the other holds.

[Organic]

Hurkyl,

It is clear as a middle-noon sun, if n exist then n+1 exist.

How can you say that this is not an induction, which its product is clearly Omega(=aleph0)?

Please look here: http://us.metamath.org/mpegif/aleph0.html

[Organic]

Matt,

You still don't get it, what you call two lists is the same list with
3 different orders.


The first is top --> bot.

The second is bot. --> top

The third is one form top, one from bot., one from top, one from bot., ...

In all of the cases we have the same 2^aleph0 01 unique members.

[Hurkyl]

It is clear as a middle-noon sun, if n exist then n+1 exist.

How can you say that this is not an induction, which it product is clearly Omega(=aleph 0)?

What does this have to do with

is the result of using the ZF axiom of infinity bulit-in induction on the power_value
of 2^0, 2^1, 2^3, ...

?


(P.S. I'm content with simply saying &omega; = N)

[Organic]

Hurkyl,

Please read: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

And see by yourself the connection.

[matt grime]

Originally posted by Organic
Matt,

You still don't get it, what you call two lists is the same list with
3 different orders.


The first is top --> bot.

The second is bot. --> top

The third is one form top, one from bot., one from top, one from bot., ...

In all of the cases we have the same 2^aleph0 01 unique members.

Priceless! So, your infinte list has two ends?

What is clear is that reading the list



...00000
...00001
...00010
...00111
...01100


that every string on the list has only finitely many elements that are non-zero. My proof holds here: let x be the string on row r, as 2^r>r for all r, it follows that after the r'th postion on the string (right to left) that every subsequent entry is zero as by definition the s'th coumn starts with 2^s zeroes.



Now the second list reading the orignal list from bottom to top (which now implies that the list is finite)

...1111
...1110
...1001
...1000


the same proof demonstrates that any string on the list only has a finite number of 0's in it. Now you are saying these lists are the same? Where is the string ...1111 on th first list? I only ask because in you new diagonal paper you say it isn't on the first list. But it is on the second, and the lists emnumerate the same elements,. namely the thing you call T?




And you say if n then n+1

n is not a statement that is true or false. It is evident you don't understand the slightest thing abuot logic and its conventions. What does it mean for n to be true?

[Hurkyl]

Theorem:

If S is a set, and subsets of S have both smallset and largest elements, then S is finite.


As a special case:

If L is a list, and each sublist of L has both a first and a last entry, then L is a finite list.

[Organic]

Matt,

My proof is not about |T|=|L| but |T|>=|L|.

in T there are:

...000...1111
...000...1110
...000...1001
...000...1000

and also there are:

...111...1111
...111...1110
...111...1001
...111...1000

You still don't understand that there is no such a thing like cardinality of infinitly many objects, because any collection of infinitly many objecsts is
a non-complete collection.

Therefore its cardinality cannot be found.

[Hurkyl]

Not every collection can be represented by a list.

[matt grime]

Originally posted by Organic
Matt,

My proof is not about |T|=|L| but |T|>=|L|.

in T there are:

...000...1111
...000...1110
...000...1001
...000...1000

and also there are:

...111...1111
...111...1110
...111...1001
...111...1000

You still don't understand that there is no such a thing like cardinality of infinitly many objects, because any collection of infinitly many objecsts is
a non-complete collection.

Therefore its cardinality cannot be found.

The wholepoint of this is that you ARE claiming |T|=|L|, how do you do this? by claiming the list for T has cardinality 2^aleph-0, that it is a complete list. It isn't as you know because of th diagonal argument.

Explain why N is not complete, in your view; cardinality does make sense for infinite sets; answer any of the fatal errors in your posts today.

[Organic]

Hurkyl,

I am talking about (...111,...000] XOR [...111,...000)
XOR (...101010,...111] XOR (...010101,...000] XOR ...

[Organic]

Matt,

The fatal error is the cardinality of infinitly many elements.


Please show me how can infinitely mant elements can be complete.

[Hurkyl]

(...111,...000] XOR [...111,...000)
XOR (...101010,...111] XOR (...010101,...000] XOR ...

(a) We've told you over and over again that this notation is nonsensical; it doesn't even come close to the standard mathematical definition, and you refuse to define it (and consistently claim that you are using it in a standard way)

(b) You're trying to make a list. What does exclusive or have to do with anything?

(c) You do realize that if A xor B xor C xor ... does NOT mean "Exactly one of A, B, C, ... are true"; it means "An odd number of the A, B, C, ... are true".

e.g.: if A, B, and C are all true, then A xor B xor C is true.

[matt grime]

Originally posted by Organic
Hurkyl,

I am talking about (...111,...000] XOR [...111,...000)
XOR (...101010,...111] XOR (...010101,...000] XOR ...


Oh, look, more misuse of logic. XOR is a logical predicate, its inputs are things that are eithert true or false. what does it mean of (...1111,...000] to be true. Hell, for that matter, what does it mean? I guess you'll post about complementary logic, even though it isn't clear why you'd do that because all the questions are about proper mathematical objects.

[Hurkyl]

Please show me how can infinitely mant elements can be complete.

(I'm guessing at what you mean by complete)


If C is a collection, then C = C, right?

Then, C is clearly a complete collection of the elements of C.

N is clearly a complete collection of the elements of N.

R is clearly a complete collection of the elements of R.

[Organic]

Ok Hurkyl,



I am talking about (...111,...000] OR [...111,...000)
OR (...101010,...111] OR (...010101,...000] OR ...

[Hurkyl]

A or B or C or ... means "At least one of the A, B, C, ... are true"

A and B and C and ... means "All of the A, B, C, ... are true"


Oh, I should also note that logical expressions may only have a finite number of terms.


That still doesn't address the issue that

(...111,...000]

has not been given a meaning comprehensible to anyone but yourself.


And it still doesn't address the issue that this seems to have absolutely nothing to do with lists. Based on your earlier posts, I might guess you're trying to say "Do something that corresponds to (...111,...000], then the something that corresponds to [...111,...000), then ..."

But I know neither what (...111,...000] is, nor what that something corresponding to (...111,...000] is.

[matt grime]

Originally posted by Organic
Matt,

The fatal error is the cardinality of infinitly many elements.


Please show me how can infinitely mant elements can be complete.


Let N be the collection of all natural numbers, it is complete in the sense that it contains all the natural numbers, and its existence is not contrary to the ZF axioms (in fact it is required by the axiom of infinity). It is a set.


Let Z be the ring defined by formally adding inverses to elements in Z and including 0. Clearly it is still a set, and is 'complete' in any reasonable sense - that is the opereations of addition and subtraction do not take one out of the set. It doesn't contain any pink elephants, but then its existence doesn't require pink elephants.


Form the fraction field, clearly this is still a set as the construction of itself proves (there are countably many elements in any equivalence class, and countably many equivalence classes. It is Q, and it is again complete in terms of algebraic operations..

perhaps you should define what you mean by complete?




Let Q be

[Organic]

Hurkyl,

If a set has infinitely many elements we cannot use the word "complete" from a quantitative point of view, because no quantity can be captured and notated by one symbol, and then can be used as a meaningful input for some mathematical system.

When infinitely many elements are forced to be notated as one concept
we are dealing with actual infinity, and no theory can use actual infinity as input, and cannot explore it.

Please see: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

[matt grime]

Originally posted by Organic
Hurkyl,

If a set has infinitely many elements we cannot use the word "complete" from a quantitative point of view, because no quantity can be captured and notated by one symbol, and then can be used as a meaningful input for some mathematical system.

When infinitely many elements are forced to be notated as one concept
we are dealing with actual infinity, and no theory can use actual infinity as input, and cannot explore it.

Please see: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf


So it is your personal meaning for complete that no one else understands? What quantity do you mean?


Anyway, why haven't you answered the questions posed to you?

Does this mean you accept that your attempts to show |T|=|L| is non-sense?

[Organic]

Matt,

Please read again pages 3 an 4 in:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

and also look again at:

http://www.geocities.com/complementarytheory/Identity.pdf

Thank you.

[matt grime]

Looking back through your posts, it seems that you mean

a set C, is not complete, if given any list (enumeration) of some (possibly all) of its elements, then there is some element of C not listed.


This is exactly the defintion of uncountable.

N is 'complete' as the trivial enumeration shows, so complete is countable? Since N exists countable/complete sets exist, and clearly you accept Cantor's diagonal argument, hence there aer uncountable sets. Which in your world would be 'incomplete'. As there already exists a word 'countable' and 'complete' hasn't been well defined by you, can I suggest this as a definition, and simultaneous ask that you stop using it 'cos complete is used in lots of situations already, and why invent another name for something already known

[matt grime]

Originally posted by Organic
Matt,

Please read again pages 3 an 4 in:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

and also look again at:

http://www.geocities.com/complementarytheory/Identity.pdf

Thank you.


Why, what are these going to answer?

Just provide a defintion of complete here, shoulc be a matter of a few lines. We don't need to reread new diagonal because there are aleraedy enough errors explained to you here that you've failed to address, why would we add more.

[Hurkyl]

If a set has infinitely many elements we cannot use the word "complete" from a quantitative point of view, because no quantity can be captured and notated by one symbol, and then can be used as a meaningful input for some mathematical system.

What does "complete from a quantitative point of view" mean?

Why not?


Anyways, one thing I've told other people who don't think cardinality appropriately captures the idea of quantity is:

Don't think of cardinality as appropriately capturing the idea of quantity.

Cardinality has a rigorous set theoretical definition, which is not "Cardinality is the size of a set."

So treat it as such. It is yet another abstract mathematical idea that mathematicians use because it happens to be useful.

The same is true about ordinality. If you don't like the idea of counting to infinity and beyond, then treat it as it really is; a useful, abstract mathematical idea.

The only reason any mathematician would think as cardinal numbers as a size or ordinal numbers as counting is because it helps the mathematician understand things. For example, for me personally, such an interpretation has given be a very good intuition about transfinite induction, allowing me to very naturally extend proofs that apply in countable cases to proofs that work in uncountable cases. (Such as the proof that every vector space has a basis)

But if you don't like to think of cardinality as size and ordinal numbers as counting numbers, then don't, because, in all technicality, cardinality is not size, and ordinal numbers are not counting numbers.

[Organic]

Hurkyl,

Can you say in simple English what is Cardinality and Ordinality to you?

Thank you.

[matt grime]

Nice idea, Hurkyl. Here's how I might do it:

for every set, S, define a symbol #S, say that #S~#T iff there is a bijection from S to T. Call the equivalence classes of this relation cardinals.

For fininte sets, we can simply set #S to be the number of elements in S, for infinite sets we pick some distinguished labels

[Organic]

For me "complete" can be used only for a finite collection, where all memebers are unique (can be clearly distinguished from each other).

[Organic]

Matt,

If #S has some unique property (not quantitative) and #T has some unique property (also not quantitative) how do you difine a bijection between #S and #T?

[matt grime]

Originally posted by Organic
Hurkyl,

Can you say in simple English what is Cardinality and Ordinality to you?

Thank you.


I suspect my answer, would be yes and no. Yes to the satisfaction of a mathematician, but no to you. As this is a mathematical defintion, you lack of understanding isn't important, because you are trying to do maths only on things that have some nice worldly explanation.

Q. What is integration?

A. Naively it is 'the area under the graph' but you can integrate things that don't have graphs. You might even cite anti-derivatives, but that is only true again in certain circumstances (when the integrand is a continuous function for instance).


THings in maths are what they do, they are their definitions. Nothing more nothing less. For simple things there might be some real world explanation. Maths would be a lot better off if this attitude were discouraged. However, arguably its students at a basic level would be worse off.

[matt grime]

Originally posted by Organic
Matt,

If #S has some unique property (not quantitative) and #T has some unique property (also not quantitative) how do you difine a bijection between #S and #T?

I'm not defining a bijection between #S and #T. They aren't sets, so I don't define functions between them.

If we presume you mean S and T as sets, it is often very hard to decide if there is a bijection between them. That difficulty doesn't stop me saying that #S~#T if there is a bijection from S and T. Computability has nothing to do with it. The bijection is purely a set theoretic statement.

The even perm group on 4 elements is not isomoprhic (as a group) to the symmetries of a hexagon, but as sets they have the same cardinality.


You say 'complete' is only meaningful for finite sets; you've not still defined complete.

[Organic]

Matt,

You use a lot the word complete, so as a mathematician you have a definition for it, so please tell me what is complete from mathematician point of view?

[matt grime]

Here I woulduse complete to mean contains all the things that it ought to, all the things it can. The usual defintion for complete as we know it.


Something is not complete here if we can demonstrate that there exists something not there that ought to be - exactly as we do in Cantor's proof for the uncountability of R.



THere are mathematical definitions for complete that are contextual - a normed space is complete if every cauchy sequence converges. There may be others but I can't think of them off hand.


If you want to define complete for sets do so. So far your only attempt is to say complete is BY DEFINITION finite. Which is a bad one.

Figured out any of the answers to the questions asked?

[Organic]

A definition for complete:

A property that depends on the existence of all its elements in one and only one collection.

[matt grime]

That isn't a definition of complete. First it doesn't say what things might be complete.

It should start: A set S is complete iff then some criteria


you've just said complete is a property, but not what tha property is. As every set S contains all its elements trivially every set has this notional complete property. And seeing as every set with at least two elements is the union of two non-empty sets, you are also being inconsistet by demanding 'only one'


So you now admit that all this time you've been arguing with Hurkly and me, you've never known what it was that you were arguing about? Right, I'm off, that really is the final straw. To be accused of not understanding your ideas when you now admit you don't know what they are...? You don't think that a little too much?

[Organic]

Matt,

Your definition is: "contains all the things that it ought to"

Is this better then?

"A property that depends on the existence of all its elements in one and only one collection."

[Organic]

Also By my definition we can clearly understand why infinitely many elements cannot be completed.

By your definiton N is complete.

By my definition N is not complete because there is no such a thing all infinitely many elements.


I have to go so, bye bye for today.

[matt grime]

Oh dear I seem to be trapped.

My working defintion for complete was one that was a best attempt to put *some* mathematical meaning there to answer your challenge to give an infinte complete set. I don't claim that that is a mathematical definition. I wouldn't chose to use complete in this sense as a mathematical defintion. And as you didn't give any definition it was the best I could come up with.

If you were to press me, I would say that in the case we are talking about, some list L of elements of a set S is a 'complete' list if it enumerates all the elements of S, that is gives a bijection between N (or some subset), the natural numbers, and S.

But I still can't decide what complete might mean for a set.

Personally I would not choose to use the word complete like this in a rigorous mathemmatical context, except to give it its usual meaning in the English language - which is what my 'contains all that it ought to' was.The best off the top of my head definition of complete as an adjective - not lacking any components.


I don't see how a set cannot be complete in this sense. What element of the natural numbers is missing from N?

And, yes I would say it was better than yours as it says what complete means, yours just says complete is a property of some objects that are ambiguously defined.

[matt grime]

Originally posted by Organic
Also By my definition we can clearly understand why infinitely many elements cannot be completed.

By your definiton N is complete.

By my definition N is not complete because there is no such a thing all infinitely many elements.


I have to go so, bye bye for today.

But you've not defined complete!

"A property that depends on the existence of all its elements in one and only one collection"

does not define complete.

Continuity depends on the existence of an epsilon, but that doesn't say what continuous means, does it?

So, simply rephrase that,

A set is complete if and only if.... WHAT???!?

it has all its elements in one and only one set?

Obviously garabage.


The set N has all its elements in the set (collection!) N, so why isn't that complete, whatever the hell that means.

"no such a thing all infinitely many elements"


EH? so all sets are finite?


I've never met anyone as obstinate about refusing to understand the bleedin' obvious. Have a medal.


I notice you are still refusing to answer any of the criticisms of your argument.

[matt grime]

Summary of the arguments so far:

The following terms have been introduced by you as mathematical terms without definiton despite being asked repeatedly:

opposite; non-linearity; complementary; symmetry-degree; fading transition; structural quantitative; information point; uncertainty; redundancy; {__}; approach; closeness; mutation.

We also have the axiom of infinity induction that is a mystery to everyone as you are the only person to have ever used this phrase (except for those asking what it means).
Then there are tautologies that are false; logical operations of 'and' etc 'defined' on inputs that are not statements that are true of false; the claim that an inequality is EQUAL to a set.
Not to mention this long argument about what complete means in the rigorous sense that you can't define, but use freely. Your *opinions* about it don't count for anything mathematically, it is only what you can prove from the existing mathematics that counts in this argument, you must use the existing definitions and conventions, you cannot attempt to claim it is inconsistent if the inconsistencies you claim are not based on statements within the system. You cannot say that current mathematical thinking is flawed because it does not agree with something you've just made up on the spot without knowing anything about the maths you claim to be talking about.
Fine, if you want to claim it doesn't do what you want then add to it, use some other set theory (ZF v ZFC) but don't tell us is wrong internally, when the only way you can do so is to use external objects.

You did define entropy as a partition of a number into equal parts.

And we've not sorted out what it is that you mean by cardinality not being defined for infinite sets. It is, at best you could argue that you think all infinite sets have the same cardinality. You've proved that to be false.
And then there is the misuse of lists....

[Hurkyl]

Can you say in simple English what is Cardinality and Ordinality to you?

Sure. I intuit Cardinality as being (a generalization of) the size of a set, and Ordinality to be (a generalization of) counting numbers.


When it comes down to rigorous work, though, I take cardinality to be something that has the property that:

Set A has cardinality no less than set B iff there is a function from A onto B. (written |A| >= |B| or |B| <= |A|, also said as "set A has cardinality at least as great as set B", or a variety of other ways)

Sets A and B have the same cardinality (written |A| = |B|) if |A| <= |B| and |B| <= |A|.

This is an equivalence relation, so we can form equivalence classes, and the things we use to represent these classes we choose to call cardinal numbers.


I'll get to ordinality later.

[Hurkyl]

Lemme take another angle at pointing things out...

Cantor's diagonal argument requires there to be a map from the set of columns onto the set of rows. (Actually, the index set for the columns and the rows is supposed to be the same set, but you can tweak the argument a bit to make it a bit more powerful)

(If you want, I can go through the argument and point out exactly where this is required)

e.g.

If we have a list that is 3 wide, and 3 tall, then there is indeed map from the set of columns (of size 3) onto the set of rows (of size 3), so Cantor's diagonal argument is guaranteed to give a sequence not in the list.

However, if the list is 3 wide and 4 tall, no such onto function exists. Thus, Cantor's diagonal argument is no longer guaranteed to give a sequence not in the list. (though it may get lucky!)


You don't account for this when you try to apply Cantor's diagonal argument to the list of all binary sequences. The columns are indexed by N, and the rows are indexed by P(N). Until you can prove that there is a function from N onto P(N), it is fallacious to apply Cantor's diagonal argument to this list.


I've kind of lost track of what you're trying to assert these days; if you're still trying to claim |N| = |P(N)|, then your argument is circular, because you're using Cantor's diagonal argument to do your proof, but Cantor's diagonal argument can't be used until you've proved it!

[Organic]

Matt,

Let us start form this definition.

A set is complete if and only if all its objects can be found.

exapmle: S={1,2,3,...} is not a complete set.

[matt grime]

Originally posted by Organic
Matt,

Let us start form this definition.

A set is complete if and only if all its objects can be found.

exapmle: S={1,2,3,...} is not a complete set.

Ah, but what do you mean by found? You give me any natural number and I bet I can 'find' it on that list, you see it's the order in which things are given, you take it to mean that the list must terminate, I take it to mean that given the list and any number (which is then fixed and finite call it n) that I can find it in approximately n steps - look at the n'th term. It changes as n changes but so what?

Your hand wavy arguments do not show that the set of natural numbers and its power set have the same cardinality as you want to claim. That claim is errant nonsense and shows you don't understand things you claim to use accurately. So what that there is no finite time algorithm for listing ALL the natural numbers, that isn't important. As with so many cranks you're confusing what can be constructed from what can be defined.


Just associating some 'property' to something does not a priori contradict anything, you would need to show that this 'property' meant that constructions used (in the Cantor argument here) would fail to work. But they don't; they DO work. Just because you don't understand the convention
that it is possible to specify an infinte set or string doesn't make you right.

[Organic]

One by One is not all.

[Organic]

A set is complete if and only if all its objects can be found.

It means that S have first an last objects.

exapmle: S={1,2,3,...} is not a complete set.

[matt grime]

Eh? You've just given me a list of the Natural numbers and challenged me to 'find' all the natural numbers. I can 'find' them simply by saying there they are, that set there you've just written down, by YOUR definition contains all the natural numbers. My usual challenge stands, tell me which natural number is not on that list, go on, just one. Or define 'find' better.

I mean, I could ask you to prove in your next post that you're not the Dalai Lama, and even tell you I@ve a simple rule that if you write X, I'll believe, you, only I won't tell you what X is, and nothing else will suffice.

You're plumbing new depths here, even by your standards.

[matt grime]

Originally posted by Organic
A set is complete if and only if all its objects can be found.

It means that S have first an last objects.

exapmle: S={1,2,3,...} is not a complete set.

So we conclude that a set is complete if and only if it is finite. AND?

[matt grime]

incidentally, what if the notion of first and last is non-sensical? I mean is the openinterval (0,1) complete in your opinion?

[Organic]

No the open interval (0,1) is not complete.

[matt grime]

What about [0,1] the closed interval, that has a first and last element in some sense.

[Organic]

In this case [0,1] is not (0,1)+ 0 and 1, but some finite collection between 0 and 1, + 0 and 1 included.

[matt grime]

Originally posted by Organic
In this case [0,1] is not (0,1)+0 and 1 but some finite collection between 0 AND 1, + 0 AND 1 included.


Good, you're at last beginning to understand things you use.

So, a set is complete iff it is finite.

Now kindly explain what the hell that has to do with Cantor's Argument?

[Organic]

Matt,


Now kindly explain what the hell that has to do with Cantor's Argument?


I answer only to polite persons.

[matt grime]

This is mathematics, not a personality contest, if you think me rude AND wrong, then what better way to correct me than to demonstrate why it is that because N is not finite that Cantor's argument is wrong.

Note, 1. Youve proved it is correct, so this is going to be a test of your consistency in mathematics.

Note, 2. I imagine you're about to wheel out the 'it makes no sense to use the word 'all' for things that aren't complete [finite]'


For 2. Why is it not permissible to say N is the set of all natural numbers? I mean if it's only because the set is not finite then I maintain, as would any mathematician, that you are not using 'all' in the context of that sentence correctly. The inequality n>n-1 is clearly true for ALL natural numbers.

N is be definition the set of ALL natural numbers. It contains all its elements. The way to show 'not all' is to find one counter example, so show me some natural number not in the set of all natural numbers.

We can talk about the set of all natural numbers, the set of all complex numbers... we cannot talk about the set of all sets obviously.

[Organic]

Dear Hurkyl,

x=2

Let us examine x and P(x)

x:

21
^^
||
vv
01<-->1
10<-->2


P(x):

1 0
2 2
^ ^
| |
v v
0 0 <--> 1
0 1 <--> 2
1 0 <--> 3
1 1 <--> 4

================================================== ==================================================

x=3

Let us examine x and P(x)

x:

321
^^^
|||
vvv
001<-->1
010<-->2
011<-->3


P(x):

2 1 0
2 2 2
^ ^ ^
| | |
v v v
0 0 0 <--> 1
0 0 1 <--> 2
0 1 0 <--> 3
0 1 1 <--> 4
1 0 0 <--> 5
1 0 1 <--> 6
1 1 0 <--> 7
1 1 1 <--> 8

================================================== ==================================================


x=|N|

Let us examine x and P(x)

x:

4321
^^^^
||||
vvvv
...0001 <--> 1
...0010 <--> 2
...0011 <--> 3
...0100 <--> 4
...

If we use the ZF axiom of infinity induction on the power_value
of each column then:

P(x):

3 2 1 0
2 2 2 2
^ ^ ^ ^
| | | |
v v v v
...0 0 0 0 <--> 1
...0 0 0 1 <--> 2
...0 0 1 0 <--> 3
...0 0 1 1 <--> 4
...0 1 0 0 <--> 5
...0 1 0 1 <--> 6
...0 1 1 0 <--> 7
...0 1 1 1 <--> 8
...1 0 0 0 <--> 9
...1 0 0 1 <--> 10
...1 0 1 0 <--> 11
...1 0 1 1 <--> 12
...1 1 0 0 <--> 13
...1 1 0 1 <--> 14
...1 1 1 0 <--> 15
...1 1 1 1 <--> 16
...

As we can see, we get an ordered collection of infinitely many 01 unique sequences with Width aleph0 and Length 2^aleph0.


Conclusions:

1) Both Width and Length are enumerable.

2) Length > Width.


Cantor's diagonal method problems:

1) It was examined on arbitrary unordered collection of 01 sequences.

2) No collection of 2^aleph0 was examined.

3) Any "missing" 01 sequence is already in the 2^aleph0 collection,
therefore it MUST NOT be added to the collection.

4) Any complete collection cannot be but a finite collection.

5) Infinitely many elements cannot be completed, therefore the cardinality of infinitely many elements is unknown and cannot be used to establish the transfinite system.

[matt grime]

Edited:


Ah, so you have stopped answering me.

Anyway, you're claiming 'the axiom of infinity induction' again.

Still you are the only person to have done this, what is it?


Clearly the object you construct has both countable rows and columns because you list them, and, ergo, they are enumerable.

At least you are consistent in your nonsense, shame you really don't understand the elementary explanations given to you as to why what you're claiming is wrong.

As is eminently clear every row in your array is eventually zero by construction, so how can it possibly have cardinality 2^aleph-0. In fact you are ASSUMING alpeh-0 equals 2^aleph-0 to prove they are equal and derive a contradiction.... Some one might think you didn't have the slightest idea about mathematics. They'd be right seeing as for the last God know's how long you've been insisting that 'no infinite set is complete'. In post 34 in this thread approx. you were asked to define that or at least explain what you meant. For the next 130 posts you were unable to provide any answer to that. Eventually after much prompting we were able to decide 'complete' means finite!

Now, how do we get you to see this claim of yours is wrong?

[Organic]

Matt,

Please read carefully what I write here.

You wrote:

This is mathematics, not a personality contest


1) For me the quality of communication between persons comes before any subject.

2) By "the quality of communication" I mean that no person in this process, using any aggressive or fanatic approach related to the examined subject.

3) I think that fruitful communication is based on cool head ,open mind, and positive approach that first of all tries to understand the other person.

4) If you can accept these basic terms of "the quality of communication" than please write your posts without “wtf”, “what the hell”, ”utter b***”, and I’ll be glad to communicate.


Yours,

Organic

[matt grime]

Fanatic? You are not fanatically pushing a completely stupid idea? Sorry, Organic, you got the polite 'please elaborate' posts. You didn't, just obfuscated and said that Hurkly and I didn't understand anything and yet you couldn't even post a defintion of 'complete' for 130 posts, and even then only with severe prompting and a lot of help.

Is it any wonder you've managed to frustrate me into these outbursts? Some people have just got fed up with even dealing with you nonsensical 'proofs'.

So, a set is complete iff finite, what has that to do with Cantor's argument that you find so wrong? Rememeber it's the maths that counts, not your opinion of what should and shouldn't be done with infinite sets.

[Organic]

Dear Matt,

A simple example to your fanatic approach.

From the beginning you did not read what I wrote about "complete" and
"all" that can be found here for example:
http://www.geocities.com/complementarytheory/GIF.pdf

More then that, by your fanatic approach you wrote:

"I am not going to read your pdf files"

If you don't read and try to understand what I write, then the only voice that you hear is your own voice.

You are in some internal war between you and your fanatic approach
that does not give you any chance to hear other person's voice, which is different from your conventions.

Let us see for the last time if you can put aside your fanatic attitude, read the two parts of:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

And give a non-emotional response to what is written there, by writing page and paragraph number to every detailed response of yours.

Thank you,

Organic

[matt grime]

Slight edit:

Let us try this:

New diagonal:

page 3. Let x=|N| [:=aleph-0]

the bit in square brackets is my addition, and it is a defintion. It is unclear what you mean here, as |N| is not a number, so the previous arguments don't apply.

'axiom of infinity induction'

please explain what this is

we know what the axiom of infinity is, we know what induction is, but what is the 'axiom of infinity induction'

why are you justified in writing 'length 2^aleph-0'
by construction the cardinality of the columns is aleph-0

You appear to claim that since the 'finite' diagrams (my term) are of width n and length 2^n that it is permissible to write aleph-0 and 2^aleph-0. This is not true. As you prove later. Look at how the finite diagrams are constructed, now rigorously define the limit of this process, the best you can come up with is that there is a natural inclusion of the diagram for x=1 into x=2 into x=3 and so on.

So the limit will be something akin to the nested subset of this. Simple exercise, show that everything so constructed is countable.

Edit: THIS IS THE MOST IMPORTANT POINT, please deal with this if nothing else in the list of criticisms.
##################$# In your case the construction shows clearly that if z is the element of the power set corresponding to position n in the list, then there are only finitnely many non-zero entries in the row. z was an abritrary element of the power set. This is a contradiction. I'm not sure how more clear we can make that.$#######################


Anyway let us ASSUME that the diagram for the infinte case does list all the elements of the power set, then you prove this assumption is wrong. That is the end of the argument. You've just shown aleph-0 is not equal to 2^aleph-0.

So in no way is it justified to CONCLUDE that there are 2^aleph-0 rows.


PAage 4.

problems with cantor's argument.

1. is false. cantor's argument is defined with any ordered subset of the power set.

2. is not relevant.

3. the enumerated rows in your diagrma clearly only have finitely many elements being non-zero so that is incorrect.

4. irrelevant, as at no point in cantor's argument do we invoke the notion of complete as you define it.

5. ditto. Demonstrate that the notion of complete as you've now defined it is in anyway relevant ot the classical cantor argument, which we will reproduce at the bottom of this post.


you present a proof on page 5 of something. you claim on page 6 t prove the opposite. where is the mathematical error on page 5's proof?


page 7 'out of the scope of cantor's definition' please explain what that means, cos you seem to imply it is to do with your inability to accept that the word all has some mathematical meaning. Why is N not the set of all natural numbers. All means in maths that there is no exception.

for all, there exists, do you know what these quantifiers are?

it is your *opinion* that all is not acceptable for infinite setes, there is no basis in mathematics for that statement as far as i can tell. or indeed any other mathematician that i've met.


**Proof of cantor's argument.

Let N be the set of natural numbers, P(N) its power set, |N|:= aleph-0, we show there is no bijection from N to P(N), ie that 2^aleph-0 is not aleph-0, if you wish.


Suppose f is an *injection* f:N-->P(N)
then list the elements of the image by their preimage

f(1), f(2),f(3)....

define a set by S by n is in S iff n is not in f(n)

S is an element of P(N), by construction S is none of the f(i), so we conclude that no injection from N to P(N) can be a bijection, hence the cardinality of N is strictly less than that of P(N)**

no mention of the word all, no proof by contradiction, sets defined simply, so why is that wrong.

for what it's worth I don't like reading your pdfs because downloading takes up bandwidth, and down loading a file from a source I don't trust (in the computing sense) isn't something I encourage people to do.

In fact, you can ignore all my criticisms of your paper, and just tell me where my maths is wrong in my proof between the **'s

[Organic]

First paper:

Let us go step by step, and write first the most important things.

1) When we have and axiom saying that: If somthing then somthing+1
then please tell me what is its result?

[matt grime]

Originally posted by Organic
First paper:

Let us go step by step, and write first the most important things.

1) When we have and axiom saying that: If somthing then somthing+1
then please tell me what is its result?

If you mean, suppose i hav a family of statements labelled P(1), P(2).. and that I know that P(n) implies P(n+1) is true and that there is some r with P(r) true, then the statements P(n) are true for all n=>r

then the answer is mathematical induction.

But, as you cite the axiom of infinity induction, you presumably must know what it is, so just tell me. and the question was for you to cite it.


preferably though, just explain why my proof of cantor is wrong, and for good measure, explain why in your construction if z is in the power set P(N) it must then correspond to row n for some n, yet clearly the row n is eventaully always zero, and thus z must have only finitely many elements in it, but z was arbitrary, contradiction.

[Organic]

Matt,

Please hold your horses, I said we go step by step so again at this stage we are talking on my first paper.

In one of your posts you wrote: ( http://www.physicsforums.com/showthread.php?s=&threadid=12942&perpage=12&pagenumber=8 )

Here is a more mathematical definition.


There is a set W that contains the empty set and if any set y is in W then the set containing the union of y and the set containing y is also in . By induction contains every finite integer.


from:

http://www.mtnmath.com/book/node53.html

So as you see ZF axiom of infinity is an induction that define aleph0.

Isn't it?

[matt grime]

No, induction is used along with the axiom of infinity to construct a set which we label the integers. Aleph-0 is then defined to be the cardinality of this set. The existence of an inductive set AND induction mean that our model must contain the integers (or somethingl like it) to satisfy the ZF axioms.

This doesn't address the issue that you used

'the axiom of infinity induction'

the last word and the first 4 are understood, we just don't know what you mean whe you put the together.

In all honesty there are bigger issues - the ones I drew your attention to above.

secondly, whatever the axiom of infintiy induction is it would appeart to allow us to state, that because

n>n-1 that aleph-0>aleph-0

that should tell you that since aleph-0 is NOT and integer you can conclude nothing about aleph-0 statements based purely on induction on the integers. induction only tells you truth or otherwise about the statement P(n) for n an integer. P(aleph-0) makes no sense.

[Organic]

|N| is the cardinality of N where N is a collection of infinitely many n's.

Cardinality simply answer the question: "How many?"

A collection of infinitely many elements has no end by definition, therefore the cardinality of such a collection cannot be found.

Modern Math tries to hold the stick in both hands:

1) Using the name "Cardinality"
2) Instead of "How many?" using "Whet magnitude?"

By this double-definition "Cardinality" is clearly meaningless.

Cardinality is known only when we dealing with a finite collection.

I do not accept Cantor's idea about the cardinality of infinitely many
objects because it is simply a self contradiction that forcing infinitely many objects to have "well-defined" cardinality.

This poor double-definition "Cardinality" is a schizophrenic creature
that can jump beyond its own head.

The only reasonable infinity defined exactly and simply by ZF axiom of infinity.

We use it and get infinitely many unique columns but then their cardinality (notated as "aleph0") is a simple and healthy creature which its exact value is unknown, and to be unknown it is OK.

Also by the same axiom we get infinitely many unique rows but then their cardinality (notated as "2^aleph0") is a simple and healthy creature which its exact value is unknown, and again to be unknown it is OK.

By this approach we avoiding the schizophrenic state of Modern Math
about infinity, can get simple and reasonable results, that does
not put aside uncertainty.

[matt grime]

So we can conclude that you cannot answer mathematically any of the charges laid against you, I would suggest.

There is no problem with defining cardinalities, except in your *opinion*. Calling cardinality size (in the sense of counting its elements) is just a figurative way to express the concept.

Again you're presuming to say that the "limit" of 2^n as n goes to infinity is 2^aleph-0 because you are trying to treat natural numbers and cardinals as the same thing.

2^aleph-0 is the cardinality of the power set of the Naturals, it is not a number which you can treat in this cavalier fashion.

It seems it is your understanding of conventions again that is lacking. Just because you can write it doesn't make it so.

In fact you can entirely stop invoking the axiom because we merely need the natural numbers, and that they are not finite.

The axiom of infinity does not define infinity, it gives the existence of an infinite set, a set which does not have a finite number of elements. You are thinking that infinity is a unique object; it is a concept. Actually, striclty speaking it means that our model must contain the integers, or a set that behaves like them.

Seeing as you do not or cannot dispute the errors in your 'proof' or provide the slightest mathematical refutation of Cantor's argument within the axiomatic ZF world shall we say the issue is closed?

[Organic]

How "nice" is your model of infinity, it is so "nice" that you don't need infinitely many elements to define it.

Actually you don't need the existence of n's in N to define infinity because infinity is a concept.

More then that, you can use a function between no-input1 to no-input2 and find meaningful results for your highly sophisticated abstract Math, that only smart mathematicians like you can understand.

So let me tell you who am I in this story.

I am the little boy who cries: "THE KING IS NAKED".

And why the king is naked?

Let us examine a better model then you give:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

No map can be found without input, the input in the above model is infinitely many intersections that define N Q and R collections.

No one of these collections is a finite collection, therefore their cardinality is unknown, but even though we can find differences between their sizes, and we don't need any transfinite system for this.

The idea that aleph0 is beyond n's only pushing the system to no-intersections state, and in this state (please look at my model) you have no input for any mathematical system.

So if you can understand my model then:

1) You can clearly see that "transfinite" system is too powerful for any mathematical system.

2) If you read the first paper in http://www.geocities.com/complementarytheory/NewDiagonalView.pdf
after you understand my model, then you can see that :

1) I do not use N members to define aleph0, so as you see I have a general view on cardinality.

2) length > width

3) length and width are both enumerable.

4) we don't need any transfinite system to define (2) and (3).

---------------------------------------------------------------------------
Another way to show that 2^aleph0 > aleph0 is the hierarchy of the building-blocks dependency of R objects in Q objects.

This dependency can be clearly shown here:

http://www.geocities.com/complementarytheory/UPPs.pdf


By the way, the reason that |N| = |Q| is trivial because:


(1/1)(1/2)(1/3)(1/4)...
\
(2/1)(2/2)(2/3)(2/4)
\
(3/1)(3/2)(3/3)(3/4)
\
(4/1)(4/2)(4/3)(4/4)
. \
.

that can be written as:

1 <--> 1 = (1/1)
2 <--> 1 = (1/2)*(2/1)
3 <--> 1 = (1/3)*(3/1)
4 <--> 1 = (2/2)
5 <--> 1 = (1/4)*(4/1)
6 <--> 1 = (2/3)*(3/2)
.
.

[matt grime]

a function without an input is not a function. Learn the maths.

i thought you wanted to deal with one point at a time in the refutation i gave. now you don't wish to deal with any of them. or learn any maths. just say it's wrong and your right.

[Organic]

"A function without an input is not a function."

I'll cherish this clever sentence for the rest of my life.

So, can you see what I see or not? (before we go to the next paper).

[matt grime]

Look, i've had a bad day, and your idiocy is no concern to me any more. Check it out, a function has a domain, it is a set, because the functions we are talking about are functions of sets.

exlpain why the proof i gave of cantor's argument is wrong. explain what the axiom of infinity induction is.

[Organic]

Dear Matt,

Go take a good night sleep, and we shall continue tomorrow.

[phoenixthoth]

matt and organic,
please read page six, paragraph 3 of
http://www.alephnulldimension.net/matharticles/tuzfcver9.pdf

cheers
phoenix

[Organic]

Matt,


**Proof of cantor's argument.

Let N be the set of natural numbers, P(N) its power set, |N|:= aleph-0, we show there is no bijection from N to P(N), ie that 2^aleph-0 is not aleph-0, if you wish.


Suppose f is an *injection* f:N-->P(N)
then list the elements of the image by their preimage

f(1), f(2),f(3)....

define a set by S by n is in S iff n is not in f(n)

S is an element of P(N), by construction S is none of the f(i), so we conclude that no injection from N to P(N) can be a bijection, hence the cardinality of N is strictly less than that of P(N)**


If you want to see what I have to say about it you can find it in the second paper here:

http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

[matt grime]

It isn't clear that your observations apply to my proof because mine isn't a proof by contradiction, the one you include is, and isn't the proof as cantor gave it, in fact mine isn't his orginal proof, it is the second one he gave. modern convention means that peopleare taught the proof by contradiction method. try relating what you think to the constructive proof above.

as for what you have written

"All members which included in S , are different from each other.
Any member of Z* can be mapped with some member of P(Z*), once and only once.
Therefore t is different from each member in S, therefore t MUST BE INCLUDED in S."

page 8

is incoherent in its English and its mathematical conclusions make no sense. It is just nonsense, sorry.

[matt grime]

Sorry to double post but moz's form filling attributes just went bezerk.

The set S is I think defined - I don't know where you got that proof from of Cantor's argument but if I were the person who wrote it and put it out in the public domain I'd be ashamed of it. I don't mean the ideas behind it but the presentation is awful.

Once more it is you not understanding how to interpret all. The set of ALL n such that n is not an element of f(n) is a set - it is a clearly defined subset of the set of natural numbers, it is the complement of the set of m with m in f(m), again a clearly defined set. I'm sorry that the level of sophistication of your mathematical abilities isn't able to cope with these things, but that doesn't stop it being true. Or do you have some bizarre set theory in mind?

"If we want to keep S as an existing member, we MUST NOT INCLUDE t in S ."

What does this mean? keep S as a member of what?

You may wish to do your 'maths' in some bizarre model of some set thoery where those things are not sets, but that is not an issue to do with the correctness of the proof in everyone else's model of their set theory.


The proof of Cantor's argument is perfectly logical.

m

[Organic]

Hi Matt,

My proof is very simple.

I clearly show that subset S that defined as:

S = {z in Z* such that z is not in f(z)} cannot exist in P(Z*) because by option 1 t must included in S, and by option 2 t must not included in S.

Cantor's point of view is problematic because it forces S definition to be the "checker" of t existence, which is simply non-logic because t can exist without S but S cannot exist without t.

As I wrote, S existence depends on objects like t, therefore we have to check S existence by t and not t existence by S, as Cantor did.

Please read again about the hierarchy of dependency in my second paper page 2.

Shortly speaking, S can exist only if t is "out of its scope", which means that S can exist iff t is not forced to be included in it.

My proof holds iff you understand and accept this point of view.

[matt grime]

it would appear that you don't understand the idea of proof by contradction. it is exactly because of the problems that happen when you define S in this way that we conclude that the function cannot be a bijection. we show there is no such S, which there would be if there were a bijection. therefore there is no bijection. end of story, i'm sorry you cannot see this but the notions of scope ond order you introduce are neither here nor there. if you don't like the contradiction, just assume f is an injection, not a bijection conclude S isn't in the image and conclude it cannot be a bijection. end. done. no issues.

[Organic]

we show there is no such S, which there would be if there were a bijection.

This is exactly my proof, which shows that S definition cannot exist without any connection to any mapping result between Z* and P(Z*).

Therefore it cannot be used to conclude anything.

[matt grime]

yes it can, it is used to conlude that as we assumed a bijection, and this led to a contradiction (a paradox, a statement that is true and false simultaneously), hence the ASSUMPTION is invalid. Find out what proof by contradiction means. this is one.

as my proof demonstrates, if you don't like contradiction, you don't have to use it. many proofs by contradiction are unnecessary just like this is.

[Organic]

S definition cennot exist because P(Z*) does not exist without Z*.

Simple as that.

[matt grime]

"I cannot comprehend the lack of understanding that would lead anyone to think that"

I think that was Babbage's response to the question asked of his calculating machine, but if you put the wrong numbers in do you get the right answer anyway?

If you don't like contradictions, then take my proof which is not a proof by contradiction.

[Organic]

Matt,

In both cases S definition cennot exist because P(Z*) cannot exist without Z*.

Simple as that.

[matt grime]

So now the integers don't exist? In that case you are not in a model for ZF, so any conclusions you draw about ZF are wrong, or at least completely unproven.

Keep digging.


Oh, you're not about to do the 'all' has no application in infinite sets thing again are you?

Round and round we go.




You didn't respond to all of the comments on your new new diagonal argument, btw.

Especially this one from where you magic up the infinity axiom of induction nonsense again. (Note, ZF does not DEFINE aleph-0, it requires there is a set of cardinality aleph-0 in your model.)

By your construction labelling the alleged 2^aleph-0 rows that you claim represent the power set of N with N we can see

##################$# In your case the construction shows clearly that if z is the element of the power set corresponding to position n in the list, then there are only finitely many non-zero entries in the row. z was an abritrary element of the power set. This is a contradiction. I'm not sure how more clear we can make that.$#######################

[Organic]

So now the integers don't exist?

How you come to that conclusion?

It is trivial that P(Z*) does not exist if Z* does not exist.

Also S={z} where S is not a new member in P(Z*) , therefore must not be added to P(Z*).


The width of 01 collection is constructed from infinitely many 01 notations, and also its length, so where is exactly your "finite" collection?

[matt grime]

Originally posted by Organic
How you come to that conclusion?

It is trivial that P(Z*) does not exist if Z* does not exist.

Also S={z} where S is not a new member in P(Z*) , therefore must not be added to P(Z*).


The width of 01 collection is constructed from infinitely many 01 notations, and also its length, so where is exactly your "finite" collection?

SO the integers do exist, and so does P(N). why did you bring up their non-existence? 2 posts ago for accuracy.

what is *my* finite collection?


please deal with the fact that i've presented you with a non-contradiction proof for cantor (there are incidentally at least three more lying around) and with the problem about your construction: if z is any element in the power set it MUST lie at position n in your list for some n. but then z can only have finitely many non-zero elements in it, contradiction if your list is an enmueration as you insist it is.

[Organic]

Look Matt,

2 posts ago I wrote:

"S definition cannot exist because P(Z*) does not exist without Z*".

1) Please show me exactly how do you come to the conclusion that there are no integers, according to this sentence?

2) Also show why the hierarchy of dependency is meaningless to you.

---------------------------------------------------------------------------

A) Each sequence in my 01 collection has infinitely many 01 notations.

B) Therefore the 01 notations of each row can be put in 1-1 and onto with N objects.

C) Therefore the cardinality of each row is |N|.

D) But because i define by induction the power_value of each column,
i get an ordered collection of 01 sequences where (and i say it again) the cardinality of each row is |N|.

E) Therefore the cardinality of the collection of these sequences is |P(N)|.

F) This ordered collection is constructed in such a way that it cannot skip or miss any 01 combination.

G) But because the cardinality of infinitely many elements cannot be written as some quantity, we can use instead the invariant of size ratio between width and length.

H) In a finite collection the ratio is given by 2^n - n = |h| where |h| is a finite number of the cases that the diagonal does no cover.

I) In infinite collection the ratio is given by |P(N)|-|N|=|H| where |H| is the domain (of infinitely many cases) that cantor's diagonal does not cover.

J) But because the building method define the uniqueness of each 01 sequence, there is a 1-1 and onto on both width and length, and Cantor's diagonal is meaningless.

[matt grime]

Shall we put the existence or otherwise of Z* down as a misunderstanding?


as for the other points

A. is an ambiguous sentence, I think you mean each row has infinitely many entries in it

B. doesn't follow from A

D induction doesn't allow you to do that.

E doesn't follow from any of the above. again you aren't using induction correctly

G the ratio between width and length is not constant in the finite case 2^n/n is not constant it is a monotone decreasing sequence.

H is about a finite case

and 'I'cannot be deduced by induction on the finite case. besides, which I really don't think you are in any position to claim to know how to do cardinal arithmetic for infinite cardinals.


so stop claiming induction tells you what the infinite case is, because that is not what induction tells you. It tells you the statement is true for an infinite number of cases, that is not the same thing.


If the rows are both countable and in bijection with the power set, let z be an element of the power set, by construction z occurs at row n, then in row is always zero reading left to rught after the n'th position - the diagram you construct is a lower triangular matrix. contradiction.

conclusion the countable set you labelled is not the power set (it is the finite power set)

if you knew the difference between coproduct and product this would be obvious. even if you don't it is still obvious, in fact.

you cannot use induction to claim things about aleph-0: it is not a number, thiis is not allowed!

counter example, 2>1, and for all n it follows n is not equal to n+1, by induction, as one is strictly greater than the other, therefore, in your system aleph-0 is not equal to aleph-0.


why is the heirachy meaningless? you mean apart from the fact that it uses undefined terms again?

[Organic]

In (G) the invariant is the formula 2^x/x.

The result is not a constant but depends on x.

You have to understand that we are talking about a paradigm change in the infinity concept, when used by Math Language, so we are not talking about technical incompatibilities, but on conceptual incompatibilities.

Today's Math does not distinguish between actual and potential infinity.

For example |N| is something which is beyond the elements that it suppose to be their measurement.

This is a qualitative change that pushing any explorable system to be too powerful for any exploration.

Please look again at this model:

http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

and show us how can we find a map between oo in this model and some collection of infinitely many objects(=intersections)?

I say that we can’t because oo is an actual infinity and any collection of infinitely many elements cannot be but a potential infinity.

Therefore |N| must be a potential infinity and only then it can be used as some meaningful input for Math language.

Because |N| is a potential infinity, it is not beyond the elements that it suppose to be their measurement.

Shortly speaking, transfinite cardinals cannot exist as useful mathematical input, and any math method that using them is not going to survive in the long run.

The one and only one way to deal with infinity is only in the scope of potential infinity, where concepts like uncertainty and redundancy are fundamental and very fruitful concepts of Math.

As for the hierarchy of dependency, this is the gate for better understanding of concepts like complexity, and maybe for the first time there is a chance to develop a comprehensive and powerful language that can develop the connections between the abstract and the non-abstract in our universe(s).

[matt grime]

Once more when faced with maths you run away and hide in philosophy. try posting there. if you want to understand what infinity is, or isn't (because you appear to have clue as to its mathematical interpretations, yes plural) then look at a recent sci-math posting

http://groups.google.co.uk/groups?dq=&hl=en&lr=&ie=UTF-8&threadm=c0fasc%24t84%241%40news.math.niu.edu&prev=/groups%3Fhl%3Den%26lr%3D%26ie%3DUTF-8%26group%3Dsci.math




and read some of the answers there

correction i should of course have said 2^n/n is monotone INcreasing.

[Organic]

Matt,

Any paradigm's change cannot be done by the conventional point of view of some system.

A lot of fundamental concepts (and in this case the infinity concept) are deeply changed.

Mathematicians like very much to send these fundamental changes to philosophy area instead of take the challenge and seriously try to examine their impact on current Math language paradigms.

You are the one who hiding here, because you refuse to show how your paradigm can deal with my new model of infinity that can clearly shows the differences between actual and potential infinity
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf .

Because I am talking about a paradigm change in the infinity concepts, I am totally aware that in the first stage there is a very big problem to understand my point of view, and we must understand that there is no way to really understand this point of view form the old point of view.

I can be sure in two things about you Matt.

You don't have the ability to see Math language from a different point of view, because your approach about Math is too emotional.

Any way, this forum is a forum of theory development where what you call philosophy is welcome, and from a very good reasons that maybe you can’t understand.

[matt grime]

Fine, keep it as a some 'intellectual' exercise, tell me that my maths can't cope with these new concepts, but don't tell me that these are extant concepts that my theory defines AND misunderstands, and don't attempt to misuse my theory to show my theory is wrong. In particular you should excise all references to induction because you are simply wrong, or state that this is your new induction, where by P(n) implies P(n+1) tells us P(aleph-0) is true, despite the fact the P(aleph-0) might not even make sense, example

for all n, 9|10^n-1

can be proved inductively, yet what does it mean to say 9|10^(aleph-0)-1?

You are free to develop whatever ideas you want, my issue is entirely with your attempts to break current mathematics by claiming things *ought* to be true in that system which clearly aren't. Your arguments involving maths are not mathematical, they do not conform to any of the standards required, therefore you shouldn't say it's [mathematics] wrong. So there is no problem with mathematics as it is understood because the things you are doing aren't doable in mathematics.

Your inductions are not valid, your deductions often make no logical sense, you cannot answer the mathematical criticisms of your work about mathematics as we understand it, you have misunderstood the ideas of proof by contradiction, invented new terms like 'axiom of infinity induction'.

You cannot claim Cantor's proof is inconsistent within the framework we operate in if you are using concepts outside of that framework. The assumptions you make particulary in your inductions are not consistent with the model of set theory most commonly used, that means that you cannot claim that theory is wrong.

If in your opinion the word 'all' can only be used with finite sets then you are doing a different kind of mathematics, and it cannot imply that the arguments within mainstream maths are wrong within mainstream maths because that *opinion* is nothing to do with maths.

In a model of set theory with a 'largest set' then Cantor's argument would be false, but it wouldn't contradict it in a model without a largest set. It would a fiortori be that the largest set's power set had to be defined differently IF it existed - sets might not have to have power sets in some models, it is an axiom of ZF that the power set of every set is a set, perhaps in some other it would be that the power set is NOT set, just like the collection of sets is not a set but a proper class in ZF.

It is not your theory that angers me, but your presumption to be able to say things about mathematics, a subject you clearly do not understand

[Organic]

Again, there is no objective thing like "Mathematics".

Therefore it can be deeply changed when its paradigms are changed.

My point of view deeply change its current paradigm about the number concept, the set's concept, the infinity concept, the continuum and discreteness concepts, and also clearly shows Math language limitations.

All these changes are simple and fundamental, and they are based on coherent models that cannot be ignored.

Also the new paradigm researches our own abilities to develop it as an important and a legal part of it.

Complementary Logic is the logical base of the new paradigm, and we can clearly show that Boolean and Fuzzy logics are private cases of it.

http://www.geocities.com/complementarytheory/CATpage.html

[matt grime]

I think coherent is the last word anyone could use for you. And will you stop using the word private like this. And as you haven't got a clue about maths as it stands, do you think you are the best person to talk about its paradigms?

[Hurkyl]

Paradigms don't change, they shift. When someone comes up with a revolutionary new way to do things or to think about things, it's just that... a NEW way to do things or to think about things.

If, indeed, you are bringing about a paradigm shift in mathematics, you do not alter any old mathematics!

[Organic]

No theoretical system can survive without being aware to its limitations.

It means that any x output can be only a model(X) input.

Shortly speaking, x=model(X).

Math is first of all a form of theory, therefore any concept that can be used by it is only a model(CONCEPT).

For example, let us take infinity concept.

If INF is infinity itself (= actual infinity) , then inf=model(INF)=potential infinity.

Please look at this model for better understanding:
http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

In this way we first of all aware to our input limitations, which are:

No input = model(EMPTINESS) = lowest limit.

No input = model(FULLNESS) = highest limit.

If we translate this to set's representation then:

{} content = model(EMPTINESS) = lowest limit.

{__} content = model(FULLNESS) = highest limit.

Between these limits ({},{__}) we can find inf=model(INF)=potential infinity, where inf has two input forms:

{.} = singleton, which is a localized element.

{.__.} = non-singleton, which is a non-localized element (connect at least two different singletons).

{.} and {._.} can appear in two basic collections:

Collection {a, b, c} is finitely many elements.

Collection {a, b, c, ...} is infinitely many elements (=inf) .

Any non-empty collection which is not a singleton, is an association between {.} and {._.}, for example:

b b
{a , a}
. .
| |
|___|_
|


{a , b}
. .
| |
|___|
|


I opened a new thead for this at:

http://www.physicsforums.com/showthread.php?s=&threadid=14416