How Does Complementary Logic Redefine Mathematical Infinity?

[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 theory 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?d...rev=/groups?hl=en&lr=&ie=UTF-8&group=sci.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?