The power of the transfinite system

[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 infinite 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 infinite 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:

https://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 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.

 

[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 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.

 

[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.