How Does Complementary Logic Redefine Mathematical Infinity?

[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.

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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?

 

[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:

https://www.physicsforums.com/showthread.php?s=&threadid=14416