The power of the transfinite system

AI Thread Summary
The discussion centers on the implications of transfinite systems, particularly the concept of aleph0, which is argued to be too powerful for any information structure based on finite bases. It posits that when reaching the power of aleph0, traditional information systems fail to yield meaningful results, leading to a dichotomy between emptiness and fullness. The participants debate the validity of these claims, questioning the definitions and interpretations of mathematical concepts presented. The discourse highlights a fundamental disagreement on the nature of infinity and the limitations of models in representing mathematical truths. Ultimately, the conversation underscores the complexity and challenges in discussing transfinite mathematics and its implications for information theory.
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  • #52
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.
 
  • #53
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
 
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  • #54
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.
 
  • #55
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 exactly 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.
 
  • #56
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.
 
  • #57
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.
 
  • #58
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.
 
  • #59
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.
 
  • #60
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.
 
  • #61
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 believe 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.
 
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  • #62
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".
 
  • #63
A Simple question:

Can a rigor proof be changed?
 
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  • #64
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.
 
  • #65
And what if basic concepts are chaneged?, for example:

1) Infitinty.

2) Addition AND Multiplication.

3) Set

4) General definition of a NUMBER
 
  • #66
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.
 
  • #67
And what if concepts like redundancy and uncertainy are used as fundamentals in our logic system?
 
  • #68
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.
 
  • #69
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?
 
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  • #70
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.
 
  • #71
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.
 
  • #72
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.
 
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  • #73
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 exactly 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.
 
  • #74
Dear Matt,

If you say that I don't 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.
 
  • #75
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.
 
  • #76
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.
 
  • #77
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.
 
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  • #78
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?
 
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  • #79
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.
 
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  • #80
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?
 
  • #81
In both cases A has the elements of B.
 

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