The power of the transfinite system

[Organic]

Hi,



First, please look at this example (It takes about 1 minute to load it) :

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


From this example we can understand that if aleph0 is related to all N members then any n of n^aleph0 cannot be but 0.

The reason is very simple: When we deal with all N members, the power of |N| (=aleph0) is too strong for any information structure, which is constructed on some n>0 base.

Shortly speaking, the "rainbow of information" does not exist when we reach the power of aleph0.

No information --> no input --> no results --> no conclusions.

Therefore transfinite universes, which constructed on ...2^(2^(2^aleph0))... does not hold.

When we try to force the transfinite idea on any information system (including Math language) we get:

...2^(2^(0^aleph0))... = 1 = {__} = Fullness.

XOR

...0^(0^(0^aleph0))... = 0 = {} = Emptiness


{} XOR {__} contents are actual infinity and cannot be reached by any information system (including Math).


Organic

[Hurkyl]

But {__} has been reached by your system, thus your system cannot have any information in it! (Otherwise it would be an information system, right?) [:)]

[Organic]

No {} and {__} are the unreachable limits of the open interval ({},{__}).

please read again this paper and its links:

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

Again:

{} is only an x-model of the content Emptyness.

{__} is only an x-model of the content Fullness.

Please look here:

http://www.geocities.com/complementarytheory/CATpage.html

and then read the Major Theorem.

Yours,

Organic

[master_coda]

Since your system can describe {___} it clearly reaches it. Thus we have two possiblities:

1. Your system contains no information.
2. {___} is not in fact unreachable by an information system.

Unless by "reach" you mean something else. But if we can describe something, we don't really care about anything else.

[Organic]

Dear master_coda,

You wrote:

...if we can describe something, we don't really care about anything else.


By your comment you do not distinguish between x-model and x-itself.

The set idea cannot be but an x-model, therefore {} is an x-model of Emptiness and {__} is an x-model of Fullness.

Any description is only an x-model, for example: any description of silence is not silence itself.

This is my Major Theorem before I starting to develop any theory.

Again, no theory is x-itself but only an x-model.

If you don't understand or don’t agree with that then we cannot communicate in this subject.

My point of view can be found here:

http://www.geocities.com/complementarytheory/CATpage.html

[master_coda]

So model of information cannot contain any information itself.

Likewise, any model of logic cannot contain any logic itself. Thus we cannot use logic to talk about logic.

Of course I cannot agree with that.

[matt grime]

At Organic's request, here are some comments on his material:





>http://www.geocities.com/complementarytheory/PTree.pdf

Pretty, but as with most of the articles there unclear.

>From this example we can understand that if aleph0 is related to all N members then any >n in n^aleph0 cannot be but 0.

How or why can one understand this? The article has an unmotivated picture with some interpretation that is not explained clearly. and what does 'n in n^aleph-0' mean? 'in' would usually indicate some kind of set were being talked about.


>The reason is very simple: When we deal with all N members, the power of |N| >(=aleph0) is too strong for any information structure, which is constructed on some n>0 >base.

what does it mean for something to be 'too strong for any information structure' and for that matter, what is an information structure?

>Shortly speaking, the "rainbow of information" does not exist when we reach the power >of aleph0.

More undefined terms. And as with a lot of problems people seem to have with infinity, how does on reach the power of aleph0. There is (reasonably explicitly) some presumption that one 'travels' towards infinity, but because one never reaches it thus having all these inherent contradictions in mathematics. Often it is to do with constructibility and Turing Machines and the issues of finitely many steps.

>No information --> no input --> no results --> no conclusions.

>Therefore transfinite universes, which constructed on ...2^(2^(2^aleph0))... does not >hold.

Obviously wrong.

>When we try to force the transfinite idea on any information system (including Math l>anguage) we get:

>...2^(2^(0^aleph0))... = 1 = {__} = Fullness.

Ok. the left hand side of that is an infinite cardinal I think, though I'm not sure what 0^aleph0 is, the next term is a finite cardinal, the next is something that we must interpret as a set, though which one is never explained, the last is a word. Do you not think that '=' is the wrong symbol to use here?

>XOR

>...0^(0^(0^aleph0))... = 0 = {} = Emptiness

I think ditto is a valid comment


>{} XOR {__} contents are actual infinity and cannot be reached by any information >system (including Math).


>Organic

Matt

[Organic]

Some correction:

I wrote:

...2^(2^(0^aleph0))... = 1 = {__} = Fullness.

XOR

...0^(0^(0^aleph0))... = 0 = {} = Emptiness

----------------------------------------------------------------------------

Instead, it has to be:

...0^(0^(0^aleph0))... = 1 = {__} = Fullness.

the second one ...2^(2^(0^aleph0))... = the "never ending" tree:

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

[Organic]

Master_coda,

You are going too far with your conclusions.

We can talk about anything in any form, depth or direction but always we have not to forget that any theory about something is never the something. that's all, nothing less, nothing more.

Logic is the simplest form of some x-model, if it was x then and only then it was beyond our power to deal with it.

Shortly speaking, we can deal and develop any form of x-model, and Complementary Logic is a good example for this.

And why it is a good example?

Because first of all it is aware to its limitations as an x-model.

[master_coda]

But the only reason your "theory" is aware of its limitations is because you've made up limitations. You haven't shown what the limits in fact are, you've just said "there are limits, and I'm making up words like emptiness and fullness to describe them".

[Organic]

Hi matt grime,

Where is your imagination?

Do you really cannot imagine that no tree of any base can carry the power of aleph0 and survive?

Again, no information can be used as input when we reach actual infinity.

[master_coda]

Organic, why do you keep using the term "aleph0"? You don't agree with any of the mathematics behind it, and you don't use it in any way resembling it's actual definition.

[Organic]

Aleph0 = {__} through my point of view.

If i want to make a mutation in this concept, i have no choice but to show my new interpretation to aleph0.

Therefore ({},{__}) = ({},aleph0).


More than that, any concept in x-model can be changed by its meaning.

[master_coda]

Whats the point of calling it aleph0 if you don't have aleph1, aleph2, etc. ?

Any concept in any model can be changed.

[Organic]

First I like the name because my language is hebrew and aleph is the first leter in my alpha-beth.

Also through my point of view aleph0 has exactly 0 points, and this is the reason why aleph0 = {___} where ___ has 0 points.

But I'll be glad to know what name to you want to give to {__}.

[master_coda]

Call it {___}. At least that'll help avoid equivocation.

[Organic]

Thank you Master_coda, I'll call it Full set (which is the opposite of Empty set).

[matt grime]

Originally posted by Organic
Hi matt grime,

Where is your imagination?

Do you really cannot imagine that no tree of any base can carry the power of aleph0 and survive?

Again, no information can be used as input when we reach actual infinity.

What the hell as imagination got to do with it? I can imagine lots of things that are false, it doesn't mean I call them mathematics.

Again you are talking about reaching infinity. This demonstrates you don't understand what infinity means. Which infinity? The infinity of [0,oo)? there is no such point. One doesn't reach it. The north pole in the standard one point compactification of the complex plane?

[Organic]

Matt,

Please look at the white arcs in: http://www.geocities.com/complementarytheory/PTree.pdf

When we reach aleph0, they are gone (become a one solid line {___}, and base=0 ).

Without them there is no information to deal with.

More than that, {___} content is too strong to be measured by 1-1 map.

Please also look at: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Thank you,

Orgainc

[matt grime]

Originally posted by Organic
Matt,

Please look at the white arcs in: http://www.geocities.com/complementarytheory/PTree.pdf

When we reach aleph0, they are gone (become a one solid line {___}, and base=0 ).

Without them there is no information to deal with.

More than that, {___} content is too strong to be measured by 1-1 map.

Please also look at: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf

Thank you,

Orgainc

How hard is it to understand that one does not physically 'reach aleph-0'. It is not a point in space, it is not even a limit point of physical space. It is the cardinality of the Natural numbers.

Further, a picture is not a proof, just because you can't draw infinitely divisible objects does not mean mathematically that after a finite number of steps repeatedly dividing by two gives you zero, for instance.

Please stop misusing mathematical ideas and claiming they are still mathematical.

What for instance does it mean to be measured by a 1-1 map? measured on what scale? Measured by what means?

You've ignored the question again about what it means to be too strong to be measured. In fact you ignore anything you can't explain, which is pretty much every criticism laid at your door. Define this term. Go on, just define it, here in this forum, and not by posting some unrelated pdf file.

[Organic]

Matt Grime,

My pictures don’t have any connections to physical sizes, they are rigorous exactly like any definition that expressed by notations.

One thing is for sure, you don’t have the ability to translate them to abstract thoughts.

Notations, pictures, and so on are only tools that help us to organize our ideas.

These pictures are rigorous because they are based on Complementary Logic:

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



But your problem is deeper then that, you do not distinguish between x-model and x-itself, (where x is infinity) and I do distinguish between them.

Please read my main website’s page:

http://www.geocities.com/complementarytheory/CATpage.html

and also:

http://www.geocities.com/complementarytheory/X-model-X.pdf

If you don’t agree with my main theorem (after you read it), then there is no use to continue our communication on this subject.

Yours,

Orgainc

[matt grime]

Originally posted by Organic
Matt Grime,

My pictures don’t have any connections to physical sizes, they are rigorous exactly like any definition that expressed by notations.

One thing is for sure, you don’t have the ability to translate them to abstract thoughts.

Notations, pictures, and so on are only tools that help us to organize our ideas.

These pictures are rigorous because they are based on Complementary Logic:

http://www.geocities.com/complementarytheory/CATpage.html




HOwever it is you who is claiming that because, in the diagram, they all become indistinguishable that something is going on. So you are requiring a phyiscal realization.



But your problem is deeper then that, you do not distinguish between x-model and x-itself, (where x is infinity) and I do distinguish between them.

Please read my main website’s page and also:

http://www.geocities.com/complementarytheory/X-model-X.pdf

If you don’t agree with my main theorem (after you read it), then there is no use to continue our communication on this subject.

Yours,

Orgainc

You still haven't adequately defined your infinity, or its model. And you still keep posting pdfs that you know I will not read on principle.

[Organic]

If you don't read my major theorem about math by principle then this is my last reply to you.

[matt grime]

The main statement is

no model of x is x.

Where x is something you've to define at a later date?

Correct.

x is a theory, its model is a model, they are distinct, clearly not the same.

Do you also accept that a set and an inequality are not the same? And are therefore not equal?

Now, are you going to answer the challenge to clearly and unambiguously state what you mean by

'too strong to be measured'

in simple text and not via some unrelated pdf?

As a guide line:

define a 'measurement'.

define 'strength' of objects in terms of this measurement.

prove that there exist things with greater 'strength' than any given 'amount' or similar.

[Organic]

Matt,


It is very simple when you understand my point of view.

It is written here:

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

If you read it we will see that there are two types of one:

type 1) one of many (any object that belongs to some collection of finitely or infinitely many objects).

type 2) ONE (an infinitely long solid line that cannot be reached by type 1 objects).

1-1 map can be used only between type 1 objects.

When we use the words 'all' or 'complete' with some collection that include infinitely many objects, we get type 2 object.

I hope you understand that there cannot be any 1-1 map between type 1 and type 2.

Therefore the 1-1 map cannot be used if we force |N|(= aleph0) to be the cardinal of 'all' N infinitely many objects.

The main property of infinitely many type 1 objects is not to be completed, therefore we cannot talk on 'all' N objects.

A collection of sperated and distingushed infinitely many objects can exist if and only if we DO NOT have 'all' of them.

You will not uderstand this if you don't read, and (I hope)
try to understend what I write here:

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

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


Yours,

Organic

[matt grime]

So you aren't going to answer the question I asked?

I'll repeat it here:

Define precisely what it means for a set to be 'too strong to be measured'


As for your actual reply, I'm afraid I don't follow what you're attempting to do.

Please define 'one' of which there are two types. (thus making 'one' a bizarre choice of name for it).

Give examples of objects which are type 1 one and type 2 one.

And you still keep posting pdf links for me despite me asking you not to. It's almost as if you don't want me to read it.

[Organic]

Dear Matt,


My rigorous examples (my models) are based on pictures.

Your interpretation that math rigorous definitions and examples MUST be based on notations (otherwise it is not abstract) is wrong.

Notations, diagrams, pictures, and so on, are only tools that help us to organize our ideas in the most simple and rigorous ways.

I use pdf because this is the simplest way to combine graphical/textual information about some abstract and rigorous model.

I believe that you know the idiom: "One good picture = 1000 words".

Now, please look again at this pdf:

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

By using Riemann's ball as our model, we find that the top line has two results.



Result 1:

Infinitely many intersections with the middle-line, where each intersection is one of infinitely many intersections.

This universe of infinitely many intersections is what I call potential infinity.



Result 2:

The top line cannot intersect itself, and when oo-line is parallel to middle-line, then top line is the ONE, or what I call the actual infinity ( or the Full set {__} ).

[matt grime]

So you can't answer my question then? It is not in the scope of your theory to explain itself and be consistent?

A picture can be a useful tool for explaining something, it is not in itself worth anymore than that as it is realized in an imperfect system

Your Riemann Limit still insists that one must reach infinity, and that our number systems are limited to lying in certain regions. Do you know about the extend number systems that exist? Infinity is a useful point in the compactification of the Real Line or Complex Plane. You are misusing it to attempt to sa something about the non compact object.


"we find the top line has two results"

please rephrase that so it is coherent.


so this unattainable point at infinity is the total set {__}?


You genuinely believe this is worth something? Guess I'll just have to be one of those pesky scholastic sceptics. It's rapidly becoming apparent (ok, it always has been) that you have no intention of attempting to explain yourself clearly. Fine, these things seem only to be important to your understanding of something. I think there is a better way of thinking, but everyone must have there own view, and it is good that you at least think about these things. However, you are attempting to use these 'imaginations' to say that mathematics, with its boolean logic is flawed because it can't handle the infinity you pick. Well, mathematics doesn't share your opinion of infinity, or at least doesn't seem to use it how you do and draw conclusions as you do. If it breaks when you think about it in your interpretation then perhaps it is your interpretation that is wrong?

So, let us get back to the point in hand.

You can't define strength, or explain why size means things can't be measured because that is not the point of your system. OK.

Go away and complete (if it can be done as it might be infinte!) your school of thought. When you write something in a forum like this that is predicated on being mathematical expect people to react neagtively, because you are using non-mathematical arguments to show why maths is wrong. That is not a sensible attitude.

[Organic]

Dear Matt,

I answered your questions, but you want the impossible to be possible, which in this case is like:

"Please explain to me what is a color, but use only black and white".

[matt grime]

No, Doron, you used the phrase

'too strong to be measured'

I would like you to explain what that means.

what does strong mean in this case?
you didn't define it, just used it
what does measure mean?
you gave no explanation of measure.

what does the phrase

'top line has two results'

mean? that is a question about the meaning of the sentence in english, not maths.

[Organic]

Dear Matt,

Please show how you check 1-1 map between a collection of infinitely many objects and infinitely long object.

[matt grime]

That question would appear to require me to accept your complementary logic stuff.

Is this near?

Let S be the set of all real numbers, this is a collection of infinitely many objects.

Let T be the Real Line, this is an infinitely long object, I think.

let f be the map f(x) = x

this is clearly injective: if f(x)=f(y) then x=y

As you stated it you need to explain what the hell you mean by your terminology again.

[Organic]

The Full set(= {__}} is constructed by exectly 0 infinitely many objects.

Therefore |{__}| = 1.

So, show me again how can you check 1-1 map between R collection and {__}.

[matt grime]

Oh dear Doron, you aren't doing very well are you?

The full set, whatever you now think that might be, contains exactly one element? What a load of garbage.

In what way did my example fail to meet our criteria?

This fullset contains exactly zero infinitely many objects?

You are aware that makes no sense as a sentence. Oh sorry, it's constructed by zero exactly infinitely many objects. Well, that also makes no sense.

You define {__} to be 'the point at infinity' of the Riemann Sphere.

Why would I want to put that in bijective correspindence with R? That is not a pointful exercise. It isn't possible, but so what? That it isn't possible is no more relevant than the fact there is no abelian group isomorphic to S_5.

Your private definitions and interpretations don't interest me, what does is your attempts to construct all the strings of 0s and 1s using the axiom of infinity. Which is totally unjustified but you do it anyway.

You're confusing sets and their elements again.

But once more you refuse to explain your terminology, and side step explaining what it means for a set to be too strong to be measured... do you think you've won any disciples by omitting this?

[Organic]

Mr. Matt,

You write to much.

...It isn't possible...

but this is the sum of it, and you right about that.

{___} content which is ONE infinitely long object, is unreachable by any collection of infinitely many objects, and it is the top limit of Math language.

Its oppsite is the "content" of {} which is the bot. limit of Math language, and it is unreachable by any content of a non-empty set.

Shortly speaking: ({},{__}):={x|{}<--x(={.}) AND x(={._.})-->{__}}.

Please read again this including all its links:

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

Aleph0 value is under the lows of propability, because no collection of infinitely many objects can reach the contents of {} or {__}.

[matt grime]

so the fact that there is no bijection between the point at infinity of the Riemann sphere and the set of all real numbers is the basis for your reasoning?

Well, Doron, one is a set, the other isn't. Or is this distinction beyond you? I'm operating within the confines of mathematics, if you wish to assign extra meanings then do so, but it isn't mathematics. That there is no bijection has no relevance. At bes one can say there is no bijection from a set with one element which we label unjustifiably oo and a set of cardinality strictly not equal to 1! Amazing. And still what does this say? Nothing.

[Organic]

Again you don't distinguish between x-model and x-itself.

{__} is x1-model of x1, where x1 is actual infinity.

{} is x2-model of x2, where x2 is the opposite x1.

One of the interesting results of this point of view is this:

Please look at the attached jpg:

http://www.geocities.com/complementarytheory/comp.jpg

Let White be Addition.

Let black be Multiplicaction.

Let Complement be Prevent AND Create.

By Complementary Logic, Addition AND Multiplication are complement operations.

[matt grime]

how do i fail to distinguish between these things?

you're side stepping the questions again!

ignoring the issues, obfuscating, inventing more notations, accusing me of being too ignorant to understand your ground breaking research... sounds like a crank to me. just consistently and systematically define all you use. if you must do it in diagrams then you must.

i see you've stopped citing the axiom of infinity now, perhaps you've realized how little that had to do with your 'work'.

so once more explain how these 01 lists must be complete. please, i love explaining why you're talking crap about them. it's so easy it's almost not worth it but just in case someone reads it and takes it seriously i must reply. note you asked me to contrigbute here.

[Organic]

01 list is not complete in both cases, which are 2^aleph0 and aleph0.

2^aleph0 > aleph0 only if aleph0 is also not completed (or uncountable if you wish).

The result (2^aleph0 >= aleph0) = {} is because standard math says that aleph0 is complete (by 1-1 and onto).

By the way the collection of 01 (infinitely long) sequencess is constructed, no 01 combination is missing (Only ...1111 excluded).

Therefore we can find 1-1 and onto between R(-1) and N objects.

But again, this is not important.

The important thing is that 2^aleph0 > aleph0 only if aleph0 is not completed because of the lows of probability.

And when any collection of infinitely many objects is under the lows of probability, no 1-1 map result is well known.

---------------------------------------------------------------------------
The right 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]

Do you know what aleph-0 is? Reading the latest post I begin to wonder. Standard math does not sy aleph-0 is complete. Standard math doesn't even say what that means.

Please elaborate on the construction of these sequences and demonstrate how using 'the axiom of infinity induction' you get the set of all 01 combinations from the finite cases. Prove that it contains all of them except 111....
Given the list as in the article, which you claim omits none of them and can be counted by putting it in bijection with base 2 expansions, where does the string of alternating zeros and ones get sent?

Every set has a 1-1 map to itself, the identity map.

I see you area adding another element into the mix with probability. But what are 'lows' of probability?

You would need to explain more clearly your proof that the rationals are countable. That is they have cardinality aleph-0. But Aleph-0 can't be used can it? Pick and mix you results eh?

[Organic]

The identity map of 1-1 of some set to itself does not hold when we deal with a collection of infinitely many objects.

Cantor himself gave this definition:

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

"A set of S elements is said to be infinite if the elements of a proper subset S' can be put into one-to-one correspondence with the elements of S."

A collection of infinitely many elements is problematic by quantitative point of view( card(S)=card(S') is a paradox ) but by its structural property it can be found as self similarity upon infinitely many scales (which is the structure of a fractal).

Now please see this pdf again (with the fractal picture in your mind):

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

So, the structural identity of an ordered collection of infinitely many objects, can clearly be shown in any arbitrary scale that we choose, but this time without any paradox.

Shortly speaking, the quantitative identity is only the shadow of the structural identity.

Through the quantitative point of view we have a paradox.

Through the structural point of view we do not have a paradox.


Probability lows in this case are very simple:

2 = (0 XOR 1)
3 = (0 XOR 1 XOR 2)
4 = (0 XOR 1 XOR 2 XOR 3)
...
n = (0 XOR 1 XOR 2 ... XOR n)
and for infinitely many objects we have also
n+1 = (0 XOR 1 XOR 2 ... XOR n+1)

Please be aware to the fact that ...1111 is not only a one missing object but an open interval of infinitely many scales.

[matt grime]

Yes a set is infinte if there exists an injection to itself which is not surjective. That does not imply that all maps must be not surjective, in fact that is trivially false as the indentity map demonstrates. In fact you then have a category in which no object is isomorphic to itself, which is a little worrying. You have misunderstood Cantor, again.

[Hurkyl]

card(S)=card(S') is a paradox

It is certainly nonintuitive, but it's certainly not a contradiction.

[matt grime]

Originally posted by Organic
The identity map of 1-1 of some set to itself does not hold when we deal with a collection of infinitely many objects.



What the hell does that mean? Any of it.

[Organic]

Through my structural point of view the meaning of identity is intuitive, simple and much more interesting then the quantitative point view, which in this case can only distinguish between (1-1) and (1-1 and onto).

Form structural point of view any map is sensitive to both structural and quantitative properties of any explored object or operation.

Therefore I can deal with information, which is beyond the horizon of standard quantitative approach.

For example, please show me the difference between multiplication and addition, by using Standard Math.

[matt grime]

The meaning of indentity might be intuitive to you, but you've not defined it anywhere have you? So what is it?

mine is Id_S(x) = x for all x in the set S

what's yours? for instance what is 'Id' on the set of natural numbers? Or just a finite set if you prefer.

The difference between addition and multiplication? Well, let's take the nxn matrices over some field for n greater than 2 addition is commutative and multiplication isn't. As you didn't restrict me to a particular addtion or multiplication that ought to do. And if you are going to define the addition and mult. would you also define what it means for them to be 'different' here. x*y is not equal to x+y for all x and y? is that enough? x=2 y=3 does that for me. and i can do it from set theory too if you want to introduce product and coproduct.

[Hurkyl]

Through my structural point of view the meaning of identity is intuitive...

Then why are you the only one here who has any idea what you mean?

[Organic]

Dear Hurkyl,

It is a good question and I'll give you the answer I gave to Matt Grime.

Your world is (0 XOR 1).

My world is fading transition between (0 XOR 1) and (0 AND 1).

Your world is a private case of my word.

I cannot translate my definitions to your world for example:

Because your logical world is limited to 2D (0 XOR 1) and my world is not limited to 2D logic, when you ask me to define my system in terms of your logical 2D word, I hope that you understand that when it is translated, her point of view is lost.

So, instead of continuing these useless replies between us, I am going to open a new thread, and the I'll ask the members to show us what is the difference between multiplication and addition by using Boolean logic.

[Organic]

Matt,

You wrote:

Yes a set is infinte if there exists an injection to itself which is not surjective.

By wolfram, one-to-one correspondence is bijection, which is injection and surjection:

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

http://mathworld.wolfram.com/One-to-OneCorrespondence.html

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

So, what you wrote is wrong.

[master_coda]

This:

Originally posted by Organic
So, what you wrote is wrong.


Does not follow from this:

Originally posted by Organic
By wolfram, one-to-one correspondence is bijection, which is injection and surjection:

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

http://mathworld.wolfram.com/One-to-OneCorrespondence.html

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



Originally posted by Organic
For example, please show me the difference between multiplication and addition, by using Standard Math.

Within the system of natural numbers, it can be shown that 3+2=5 and that 3*2=6. If multiplication and addition are not different then they should always produce the same result. Thus multiplication and addition (of natural numbers) are clearly different.

[Organic]

master_coda,

I don't unerstand the firs part of your post.

About Abuot addition and multiplication please look at:

http://www.physicsforums.com/showthread.php?s=&threadid=12783

Thank you,

Organic

[master_coda]

Originally posted by Organic
master_coda, I don't unerstand the firs part of your post.


I was simply stating that providing links to definitions does not prove matt_grime wrong.

In fact, those links state something that was equivalent to what matt_grime said (about a set being infinite if and only if there exists a non-surjective injection from the set onto itself). So you actually proved him right.

[Organic]

Dear master_coda,



http://mathworld.wolfram.com/InfiniteSet.html
A set of S elements is said to be infinite if the elements of a proper subset S' can be put into one-to-one correspondence with the elements of S.


http://mathworld.wolfram.com/One-to-OneCorrespondence.html
"A and B are in one-to-one correspondence" is synonymous with "A and B are bijective."

http://mathworld.wolfram.com/Bijective.html
A map is called bijective if it is both injective and surjective.



Conclusion: Set S is infinite iff it is bijective to a proper subset of itself.

(Because of this conclusion any identity map of set S to itself is a paradox form quantitative point of view, when S is a collection of infinitely many objects.)

And you wrote:

In fact, those links state something that was equivalent to what matt_grime said (about a set being infinite if and only if there exists a non-surjective injection from the set onto itself). So you actually proved him right.

Please explain how I actually proved him right?

Thank you.

Organic

[master_coda]

Originally posted by Organic
Conclusion: Set S is infinite iff it is bijective to a proper subset of itself.

If you have a bijection between a set and a proper subset of said set, then you have a non-surjective injection from the set into itself.

The definition you mentioned is talking about maps of the form f\colon A\rightarrow B where B\subset A. The second definiton (matts) is talking about maps of the form f\colon A\rightarrow A.

If you have a map of the first form which is a bijection, then you also have a map of the second form which is a non-surjective injection. Thus the first definition (mathworlds) is equivalent to the second definition (matts).


Also, none of these definitions have anything to do with an identity map, so these conclusions cannot possibly be involved with a paradox with the identity map. The identity map is a bijection between a set and itself NOT a bijection between a set and a proper subset of itself.

[Organic]

master_coda,

I am not talking about the second (f: A --> A) I am talking about the meaning of being a collection of infinitely many objects.

So, when A is a collection of infinitely many objects,
its identity map (f: A --> A) = (f: A --> B) , where B is a proper subset of A.

But this is exaclty what I clime about the paradox which appears contrary to expectations, and if you read this http://www.geocities.com/complementarytheory/Identity.pdf
I am sure that you will understand my argument.

[master_coda]

You don't actually provide any paradox. You just state the definition of an infinite set and then say "this is intuitively a paradox". I don't care what you think is intuitively true. When doing math, I don't even care what I think is intuitively true.

[Organic]

master_coda,

You miss fine details about my argument.

First please read by yourself what is the full meaning of the word paradox:

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

We can't ignore any part of what is written there.

So, if from some point of view there is no paradox at all, then this point of view is better then another point of view, which is against our simple expectations.

[master_coda]

Originally posted by Organic
master_coda,

You miss fine details about my argument.

First please read by yourself what is the full meaning of the word paradox:

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

We can't ignore any part of what is written there.

So, if from some point of view there is no paradox at all, then this point of view is better then another point of view, which is against our simple expectations.

Except that the paradox you describe is actually this:

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

The fact that something in math seems contradictory to you is of course, irrelevant. You have to produce an actual contradiction.

[Organic]

Again you jump to far.

Think simple (where simple not= trivial).

The basis of Math stands on at least 3 legs.

1) Logic leg.

2) Formal leg.

3) Intuition leg.

We can't ignore any of them.

[master_coda]

Originally posted by Organic
Again you jump to far.

Think simple (where simple not= trivial).

The basis of Math stands on at least 3 legs.

1) Logic leg.

2) Formal leg.

3) Intuition leg.

We can't ignor any of them.

Actually, we can ignore number 3. Intuition is entirely subjective. If we allow proof by intuition, then math loses any objective value that it has. You can assert that something is true because it is intuitive to you, and I can assert that the negation is true because it is intuitive to me. Thus allowing the use of intuition allows us to easily generate contradictions, so it must be abandoned.

Intuition is nothing more than a heuristic. It is in use for practical purposes of survival in the world. It allows humans to make quick decisions which are usually true. But it isn't always correct, since it's only an approximation.

[Organic]

Here we come to the main point.

Without Intuition leg you are a dead man baby.

We will not survive the next 5 days without it.

So, I belive it is not so logic to be dead and make Math.

Who said that redundancy, uncertainty, approximation are not natural and fundamental parts of Math.

Some of our main Axioms are based on intuition.

For example: ZF Axiom of infinity.

Please define objective-value.

I can add more, but first please answer to the above.

[master_coda]

The fact that intuition is important for survival does not mean that we should force every facet of our lives to follow intuition.

The axioms of set theory are used because they generate the results that we want. The fact that they seem true (or perhaps not) is not in fact relevant.

For hundreds of years math was weakened by intutition. Many mathematicians refused to use concepts such as "zero", "negative numbers", "complex numbers", "non-Euclidean geometry", etc. People felt that these concepts were intuitively absurd. And one by one these concepts came into common use when people realized that intuition was wrong.

Math is objective in the sense that you can mechanically check the validity of a proof. This allows us to define universal methods for deciding what is correct and what isn't. If we use intiution to decide correctness then we can no longer do that.

Of course, the correctness of a proof says nothing more than "by the rules of mathematical logic, this proof is correct". But by allowing intuition, we don't even have that.

Your complaints that math doesn't do this or that is like complaining that math won't make you a sandwich for lunch. It isn't supposed to do these things. You're complaining that math is unable to do things which is was never supposed to do.

If you want philosophy, go do philosophy. Don't whine that you want to be a mathematician but don't want the burden of basic mathematical concepts such as "formalism" and "rigor".

[Organic]

A Simple question:

Can a rigor proof be changed?

[master_coda]

Originally posted by Organic
A Simple question:

Can a rigor proof be changed?

You can create a new, modified version of a proof. But the old proof is always there as well. You can't "change" a proof in such a way the the original version somehow disappears.

[Organic]

And what if basic concepts are chaneged?, for example:

1) Infitinty.

2) Addition AND Multiplication.

3) Set

4) General definition of a NUMBER

[master_coda]

The basic concepts (i.e. axioms and definitions) are an intrisic part of a proof. So changing any of them produces a new, different proof.

If these changes are done in such a way that none of the properties the proof depend on are changed, then the new proof is also valid. For example, if you change the definition of multiplication but not the definition of addition, then proofs that depend only on addition will still be valid. But proofs that depend on multiplication are not.


However there is a very important I must make: any changes you make to definitions and axioms do not in any way affect the old proof with the old definitions. Providing a new definition of what a number is does not affect the validity of proofs made with the old definition.

For example if you provide a new definition of number where multiplication is non-commuatative, it does not mean that proofs that depend upon mulitplication being commutative are now wrong. It just means that you can't use those proofs in your new system.

[Organic]

And what if concepts like redundancy and uncertainy are used as fundamentals in our logic system?

[master_coda]

Originally posted by Organic
And what if concepts like redundancy and uncertainy are used as fundamentals in our logic system?

Well, that depends. If by changing logic you mean that you want to use different rules of inference and different truth-values and such, that can be done in a valid way. pheonixthoth was attempting to do that in some of his threads in General Math.

On the other hand, if you mean to change logic in such a way that we can no longer apply any rules of inference at all, then it would be difficult to argue that you're doing math. Kind of like trying to write a story without an alphabet or language.


However, the same catch still applies. If you invent a new system of logic and develop math in the new system, it does not affect the validity of the old system. Providing a model of math using logic with uncertainty will not invalidate math done using logic without uncertainty.

[Organic]

I agree with you that in some cases the old system is valid.

But what if it becomes a private case of a more general mathematical system?

For example Euclidean Geometry in some cases is a private case of a Non-Euclidean Geometry, or commutative multiplication is a private case of noncommutative multiplication?

[master_coda]

Originally posted by Organic
I agree with you that in some cases the old system is valid.

But what if it becomes a private case of a more general mathematical system?

For example Euclidean Geometry in some cases is a private case of a Non-Euclidean Geometry, or commutative muliplication is a private case of noncommutative muliplication?

If the new system includes the old system inside it, it doesn't necessarily mean the new system is more valid. For example ZFC set theory includes all of the results of ZF set theory as well as additional results that can be derived from adding the axiom of choice. But this doesn't mean that ZFC is more correct just because it's more general.

Also, you can't derive results in ZFC and say that they must therefore be true in ZF because they're true in ZFC. If the result you derived in ZFC depends on the axiom of choice being true, then that result doesn't hold in ZF.

If a new form of logic is more powerful than traditional logic, that doesn't mean the traditional logic no longer correct, or is wrong somehow. It just means that it generates weaker results.


Remember, more power in a logical system isn't always a good thing. Naive set theory is more powerful than ZF set theory, but the naive theory is a much worse theory, because that additional power allows you to construct contradictions. Don't make the mistake of thinking that a more powerful theory is necessarily better than a weaker one.

[Hurkyl]

One additional comment; if a new system includes an old system inside it, then all of the results of the old system must also be results of the new system.

[Organic]

I like your attitude about being careful when we try to develop a new logical system.

Therefore my basic attitude is to find ways to associate between opposite concepts in such a way that at least they do not contradict each other.

And after that we can check if they can associate and define more interesting results then the state of not being associated.

This is the main idea of Complementary Logic.

Because I am not familiar with the standard formal mathematical notations form one hand, and I did not find any existing model in pure Mathematics from the other hand, I had no choice but to write my ideas in the best way I can, which is not an easy task for professionals to understand it, and I am aware of it.

One of the things that I cared about was to use the simplest possible way to organize my ideas.

For example, I associated in a coherent way between concepts like redundancy, uncertainty and symmetry to construct a very simple model of symmetry break levels.

Then I have found that addition and multiplication are complement operations, and there is a beautiful and simple way to order them when using their complement associations on each other.

My non-formal paper with some examples can be found here
(Hyrkyl helped me to write the first 9 lines of it):

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

I am here in this forum to share my ideas with you, and learn from your experience, remarks and insights.

I think no man's work can really be done alone, and maybe one of the most beautiful and meaningful (and also powerful, therefore dangerous) languages, which is Math language, has to be developed by team work.

[matt grime]

Originally posted by Organic
master_coda,

I am not talking about the second (f: A --> A) I am talking about the meaning of being a collection of infinitely many objects.

So, when A is a collection of infinitely many objects,
its identity map (f: A --> A) = (f: A --> B) , where B is a proper subset of A.

But this is exaclty what I clime about the paradox which appears contrary to expectations, and if you read this http://www.geocities.com/complementarytheory/Identity.pdf
I am sure that you will understand my argument.

No, organic, you are wrong, that is not what is meant at any of those Wolfram links you posted. You evidently don't understand the maths here.

[Organic]

Dear Matt,

If you say that I dont understand the math here, then first you have to show that you understand my point of view, and only than you can show what is wrong in this point of view and how we can correct it.

Please correct me.

[matt grime]

I'm not talking about your theory, I'm talking about the correctness of your view on Cantor's criterion for being an infinite set.

Ok, you say that a set is infinite if

the identity map Id:A ---> A

is EQUAL to a bijective map f:A ---> B

for B a PROPER a subset.

Now, maps are surjective onto their image, the image sets are not EQUAL, therefore the maps cannot be EQUAL. If you dispute this then you are not using the correct definition of EQUAL.

Alternative proof. If f: A --> B where B is a proper subset of A, then there is some x in A not mapped to x, otherwise the image is not a proper subset. however, the definition of the identity map is that Id(x) = x for all x, so the maps are not equal.


This is not using your theory, this is to do with you not understanding mathematics as almost everyone else does. I'm not touching on your point of view in the slightest.

[Organic]

Matt what you wrote is clear and beautiful.

Can you show some interesting results that are based on the difference between these non-equal maps?

Thank you.

[matt grime]

Yes, how about Rickards criterion for Epaisse subcategories:

If T is a traingulated category, and S a ful triangualated subcategory closed under arbitrary coproducts, then it is closed under taking summands (akin to the eilenberg maclane swindle)
+ denotes direct sum

Let X be in S if X = Y + Z in T, form the infinite direct sum

Y+Z+Y+Z+Y+Z.... call this A. A is in S by construction and is isomorphic to the infinite direct sum of X with itself, which is clearly isomorphic to

Z+Y+Z+Y+Z.... call this B.

there is then a natural map from A to B by shifting components one to the right

as S is closed under triangles, the third corner must be in S, but this is just Z. Hence X=Y+Z in S too.

the shift map is the equivalent of the bijection to a proper subset.

If you want a baby version, just take the left and right shift operators on L^{infinity}

right then left shift is the identity, left then right isn't, so there is a map with a left inverse which is not a right inverse and vice versa.

[Organic]

Please give another intersting result that using this difference between left and right shifts, thank you.

Edit 1:

I have an idea about this non-symmetric shift.

When we deal with the identity map we don't care about some possible difference that can be between each pair included in the map.

Shortly speaking, in any identity map pairs_possible_difference = 0

In a collection of infinitly many objects A, for any bijective map
between A to some proper subset B of it (or some arbitrary unordered collection of A) pairs_possible_difference > 0 .

Where can I find some paper that deals with what I call pairs_possible_difference?

Edit 2:

I have another idea based on the difference between

pairs_possible_difference = 0
XOR
pairs_possible_difference > 0

Multiplication and Addition are the same only when pairs_possible_difference = 0.

What do you think?

[matt grime]

Originally posted by Organic
Please give another intersting result that using this difference between left and right shifts, thank you.

Edit 1:

I have an idea about this non-symmetric shift.

When we deal with the identity map we don't care about some possible difference that can be between each pair included in the map.


Beg your pardon, this is nonsense again.


Shortly speaking, in any identity map pairs_possible_difference = 0

In a collection of infinitly many objects A, for any bijective map
between A to some proper subset B of it (or some arbitrary unordered collection of A) pairs_possible_difference > 0 .



what does it matter if its ordered or not? the set might not even be ordered



Where can I find some paper that deals with what I call pairs_possible_difference?


I have no idea because its something you just invented, and I haven't got a clue what you mean by it


Edit 2:

I have another idea based on the difference between


Oh bugger, I've not given you more things to ruin, have I?


pairs_possible_difference = 0
XOR
pairs_possible_difference > 0



why do you have this obsession with XOR all the time. why don't you use more words?


Multiplication and Addition are the same only when pairs_possible_difference = 0.

What do you think?

What I think isn't fit for a public forum right now.

[Hurkyl]

Shortly speaking, in any identity map pairs_possible_difference = 0

In a collection of infinitly many objects A, for any bijective map
between A to some proper subset B of it (or some arbitrary unordered collection of A) pairs_possible_difference > 0 .

Are you talking about the elements of A that are not in B?

More precisely:

Let A be a set.
Let f be a 1-1 map from A into itself.
Define B to be the range of f, so that f is a bijection from A to B, and B is a subset of A.

Are you trying to talk about the elements that are in A but are not in B?

[Organic]

In both cases A has the elements of B.