Cont: Deeper than primes - Continuation 2

[jsfisher]

This is not the case with ZF(C) Axiom Of Infinity as the basis of infinite axiom set A, since no member is missing by this axiom (or more precisely, no Godel number which is used to encode G, is missing).

Huh? The Axiom of Infinity at no point says that "no member is missing" unless you are trying to observe the trivial "a set contains ever member it contains."

Or, per chance, are you trying to say something about the set of all possible statements in ZFC (or ZF if you prefer)? If so, what of it? Yes, that set would contain all possible statements in ZFC, none missing, but so what?

Or, possibly, are you trying to construct a set of all the statements that are neither provable nor disprovable in ZFC? Good luck with that since it is not a set in ZFC.

Or,...


Care to provide some clarity?

 

[doronshadmi]

Yes, of course the set of all possible sentences is complete and inconsistent. GOTO 3316 and don't come out of the loop until you understand that.

Or, possibly, are you trying to construct a set of all the statements that are neither provable nor disprovable in ZFC? Good luck with that since it is not a set in ZFC.

Or,...


Care to provide some clarity?

As jsfisher suggested, let's focused on the, so called, difference between cardinal and ordinal numbers.

All G additional statements are encoded by Gödel numbers, where (if I am not wrong) using all the natural numbers is enough in order to encode every possible G type statement.


In order to first clearly see that there is bijection between all the natural numbers and some collection of Gs type, we are using an ordered mapping, for example:

1 --> G₁
2 --> G₂
3 --> G₃
4 --> G₄
5 --> G₅
6 --> G₆
...

At this stage order is important, so the mapping is done among the terms of two sequences and not among two sets, since order is insignificant in case of sets.

After using order, we can get rid of it and remain only with the bijective map and the cardinality (the size) that is involved with it (now we deduce in terms of bijection among two sets).

Let (G₁, G'₁, G₂, G'₂, G₃, G'₃, ...) be a sequence of G type statements.

By not ignoring order (at least at the first glance) we have bijection among the terms of the two following sequences:

1 --> G₁
2 --> G'₁
3 --> G₂
4 --> G'₂
5 --> G₃
6 --> G'₃
...

By ignoring order we remain only with the bijective map and the cardinality (the size) that is involved with it (now we deduce in terms of bijection among two sets).

The cardinality of the two considered bijective maps is the same, but the output side of the second bijective map is involved also with G'_n statements, which are not found in the first bijective map.

By following this reasoning, one may conclude that by using order in the first glance, new Gs type are always added even if the cardinality remains the same.

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

But is this conclusion logically valid? Let's carefully check it.

Since cardinality remains the same in both bijections, we must not take the two considered cases as two different cases.

On the contrary, the identity between the cardinalities is used like symmetry under addition, and by using the notion of actual infinity in terms of Platonic realm, nothing is actually added in the output side of the bijective maps, since in a Platonic realm all possible Gs are already outputs of all the natural numbers.

So every mathematician that agrees with the notion of completed infinite sets/sequences in terms of Platonic realm (at least as I understand it) must agree that by using ZF(C) Axiom Of Infinity in order to establish the natural numbers and the infinite axiom set A (where A is established by using ZF(C) Axiom Of Infinity on ZF(C) itself) no possible G type statement is left out.

As a result, A must be complete and therefore inconsistent, according to Gödel's First Incompleteness Theorem.

 

[doronshadmi]

As far as I know, Gödel was a Platonist (he agreed with Actual infinity in terms of Cantor) and his main motivation behind his Incompleteness Theorems was to logically demonstrate that formal systems that are strong enough in order to deal with Arithmetic, can't be complete AND consistent, and they are also can't prove their own consistency (which means that many "interesting" formal systems are insignificant in order to deal with Platonic realms).

But if my conclusions are logically valid, then Gödel's Incompleteness Theorems also prove that the very notion of Actual infinity in terms of Platonism (which is also Actual infinity in terms of Cantor) does not hold logically (at least in the strong sense).

 

[jsfisher]

...In order to first clearly see that there is bijection between all the natural numbers and some collection of Gs type...

"Some collection of Gs types"? Are you trying to refer to the set of all statements in ZFC that can neither be proven nor disproven in ZFC? Such a set does not exist in ZFC.

 

[doronshadmi]

"Some collection of Gs types"? Are you trying to refer to the set of all statements in ZFC that can neither be proven nor disproven in ZFC? Such a set does not exist in ZFC.

Let's see.

1. Godel's Incompleteness Theorems (GIT) apply to ZF and ZFC.
2. Let A₁ be the set of axioms that define ZFC. (We only need one for the exercise. I pick ZFC.)
3. Assume ZFC is consistent (otherwise it wouldn't be all that useful). It must therefore be incomplete according to GIT. (Notice the setup for a proof by contradiction.)
4. GIT can be used to construct a statement, G, in ZFC that cannot be proven in ZFC nor can its negation be proven.
5. Let A₂ be all of A₁ plus G. That is, we are constructing a new version of ZFC in which G is an additional axiom.
6. This new version of ZFC still must be incomplete.
7. GIT can be used to construct a statement, G, in this extended ZFC that cannot be proven in the extended ZFC nor can its negation be proven.
8. Let A₃ be all of A₂ plus G. That is, we are constructing a new version of the extended ZFC in which G is an additional axiom
9. Lather, rinse, repeat.

So, jsfisher, all Gs are in the extended ZF(C) (the infinite set of axioms, called A) if infinity is taken in terms of the Cantorian actual infinity about sets/sequences, which is Platonic by nature (for example: The infinite set of all natural numbers exists in a Platonic realm as a complete mathematical object , independently of us, the human beings, and so is set A by the same Platonic/Cantorian reasoning).

The Axiom of Infinity at no point says that "no member is missing"

You know very well that by the Platonic/Cantorian reasoning ZF(C) Axiom of Infinity establishes an already completed infinite set, so no stopping-points of any kind are used in order to say something (where in your case "The Axiom of Infinity at no point says that ..." simply does not exist in case of Platonic/Cantorian reasoning about infinite sets/sequences).

 

[jsfisher]

So, jsfisher, all Gs are in the extended ZF(C) (the infinite set of axioms, called A)...

The steps I outlined generate a sequence of sets. They do not generate a set of all members in the sequence. If nothing else, you cannot satisfy the axiom schema of restricted comprehension for such a set.

 

[doronshadmi]

The steps I outlined generate a sequence of sets.

There are no steps in case of Platonic\Cantorian actual infinite sets\sequences.

All the members (in case of sets) or terms (in case of sequences) are already exist, such that infinite sets\sequences are taken as complete mathematical objects.

 

[jsfisher]

There are no steps in case of Platonic\Cantorian actual infinite sets\sequences.

All the members (in case of sets) or terms (in case of sequences) are already exist, such that infinite sets\sequences are taken as complete mathematical objects.

Without a well-defined membership function, you cannot construct a set from the sequence.

 

[doronshadmi]

Without a well-defined membership function, you cannot construct a set from the sequence.

The use ZF(C) Axiom of Infinity (Platonist Infinity) on ZF(C) itself, is exactly the well-defined membership function.

If statement x is in A then statement {x} (a G type statement) is in A. Sounds familiar?

x --> {x} bijection is complete in terms of Platonic Infinity.


This is exactly how math works if Platonist infinity is involved with recursion among a given axiomatic system that is strong enough in order to deal with arithmetic (for example: PA or ZF(C)).

Moreover, by using the ZF(C) Axiom of Infinity (Platonist Infinity) on ZF(C) itself, A axiomatic system as a complete system (and therefore inconsistent by GIT) is self evident truth.

 

[jsfisher]

The use ZF(C) Axiom of Infinity (Platonist Infinity) on ZF(C) itself, is exactly the well-defined membership function.

If statement x is in A then statement {x} (a G type statement) is in A. Sounds familiar?

For all x, it needs to be decidable if x is in A or not. The or-not part is the problem.

 

[jsfisher]

Doronshadmi,
Before we go any further, you need to be a bit clearer about your terminology. You have been vague about your set, A, and your all G-type statements.

I will try to help with the following, slightly reformed steps to generate my A_i sequence.

1. We'll use ZFC as our reference set. Call it Z₁ and it's set of axioms, A₁.
2. We will assume Z₁ is consistent, so, by Godel's Incompleteness Theorem, GIT, we know it to be incomplete.
3. From GIT we can identify a statement, G₁, in Z₁ that is undecidable in Z₁.
4. Define a new set theory, Z₂ with its set of axioms being A₁ U {G₁}. (That is, make G₁ decidable in Z₂ by introducing it as an axiom in Z₂.)
5. Z₂ must also be consistent, so from GIT we can identify a statement, G₂, in Z₂ that is undecidable in Z₂. (G₂ is also undecidable in Z₁ as it works out, too.)
6. Define a new set theory, Z₃ with its set of axioms being A₂ U {G₂}.
7. Z₃ must also be consistent, so from GIT we can identify a statement, G₃, in Z₃ that is undecidable in Z₃.
8. Define a new set theory, Z₄ with its set of axioms being A₃ U {G₃}.
9. Z₄ must also be consistent, so from GIT we can identify a statement, G₄, in Z₄ that is undecidable in Z₄.
10. . . .

Out of that, there is a sequence of Z_i's, A_i's, and G_i's. Each Z_i is nothing more than just a name for the set theory corresponding to the axioms in A_i. Each G_i is a statement in the theory that cannot be decided.

A couple of notes:

1. Each and every A_i is an infinite set of axioms. Even A₁ is an infinite set.
2. Neither of the sequences, A_i or G_i, form a set since there is no well-defined membership function. (The Axiom Schema of Restricted Comprehension, in particular, is a problem.)

Ok, so what, now, are your "A, infinite set of axiom" and your "all G-type statements"? And, once you have clarified those two things, what is your basis for the assertion that if x is in A, then so is {x}?

 

[doronshadmi]

For all x, it needs to be decidable if x is in A or not. The or-not part is the problem.

Let's demonstrate it exactly in terms of ZF(C) Axiom Of Infinity.

Since A is actually an extended version of ZF(C) by ZF(C) Axiom Of Infinity (where Infinity is taken in terms of Platonic Infinity (which means that by this axiom there exists a complete collection of infinitely many things)) then whenever a given x statement is a member of A, the set formed by taking the union of x with its singleton {x} (written as xU{x}) is also a member of A.

Now, there are at least two ways to define Gs type statements within A: explicitly (in terms of xU{x}) or implicitly (in terms of {x}).

So, whether x --> {x} implicit bijection or x --> xU{x} explicit bijection is taken, in both cases it is done in terms of Platonic Infinity, which means that A is complete and therefore inconsistent by GIT.

 

[jsfisher]

Let's demonstrate it exactly in terms of ZF(C) Axiom Of Infinity.

Since A is actually an extended version of ZF(C)

You still need to be clear what this set, A, actually is. "An extended version of ZF(C)" isn't very clear. There are many. Which one did you have in mind?

...by ZF(C) Axiom Of Infinity (where Infinity is taken in terms of Platonic Infinity (which means that by this axiom there exists a complete collection of infinitely many things)) then whenever a given x statement is a member of A, the set formed by taking the union of x with its singleton {x} (written as xU{x}) is also a member of A.

The Axiom of Infinity says nothing of the kind. It speaks only of the existence of one set that has the property you describe (of which the von Neumann ordinal is the minimal example of such a set). There is no requirement other infinite sets also have that property.

In fact, I can use the Axiom of Infinity and the Axiom Schema of Restricted Comprehension to construct from the von Neumann ordinal an identical set, but with the single member, { ∅, {∅} }, omitted. The constructed set contains {∅}, but that member would not have the successor member you claim must be there.

Now, there are at least two ways to define Gs type statements within A: explicitly (in terms of xU{x}) or implicitly (in terms of {x}).

Start by defining A.

 

[doronshadmi]

You still need to be clear what this set, A, actually is. "An extended version of ZF(C)" isn't very clear. There are many. Which one did you have in mind?

That is extended by ZF(C) Axiom Of Infinity.

The Axiom of Infinity says nothing of the kind.

In that case, Infinity is taken in terms of Potential Infinity (endless process).

In fact, I can use the Axiom of Infinity and the Axiom Schema of Restricted Comprehension ...

A is established by using only ZF(C) Axiom Of Infinity.

Start by defining A.

It is axiomatically taken as self evident truth, where Infinity is taken in terms of Platonic Infinity (which means that by this axiom there exists a complete collection of infinitely many things, for example: The infinite set of all natural numbers).

So, A is complete and therefore inconsistent by GIT.

 

[caveman1917]

It is axiomatically taken as self evident truth, where Infinity is taken in terms of Platonic Infinity (which means that by this axiom there exists a complete collection of infinitely many things, for example: The infinite set of all natural numbers).

So, A is complete and therefore inconsistent by GIT.

It might help you to think of the natural numbers under Godel numbering as statements about natural numbers instead, that is, after all, what is being encoded here. So yes, the set A containing every possible statement about the natural numbers will be complete (in both the Cantor and the Godel sense) and inconsistent since it will at least contain the set {"1 + 1 = 2", "1 + 1 = 3"} as a subset.

 

[jsfisher]

Ok, enough.

Doronshadmi continues to be unable to define anything, and he refuses to understand some really fundamental things in Mathematics.

Responding to someone who only repeats the same gibberish with no interest in learning is of no benefit to anyone.

 

[doronshadmi]

It might help you to think of the natural numbers under Godel numbering as statements about natural numbers instead, that is, after all, what is being encoded here. So yes, the set A containing every possible statement about the natural numbers will be complete (in both the Cantor and the Godel sense) and inconsistent since it will at least contain the set {"1 + 1 = 2", "1 + 1 = 3"} as a subset.

You can make it simpler.

SInce A is equivalent to ZF(C) in terms of ZF(C) Axiom Of Infinity, such that Infinity is taken in terms of Platonic Infinity, also {"1=0"} holds in A, because A is complete exactly as, for example, the infinite set of ALL natural numbers is complete.

So A (an extended version of ZF(C) by ZF(C) Axiom Of Infinity, where Infinity is taken in terms of Platonic Infinity) is complete and therefore inconsistent by GIT.

 

[doronshadmi]

Ok, enough.

Doronshadmi continues to be unable to define anything, and he refuses to understand some really fundamental things in Mathematics.

Responding to someone who only repeats the same gibberish with no interest in learning is of no benefit to anyone.

Ok, enough.

jsfisher wishes to add more objects to A, which is equivalent to the infinite set of all natural numbers.

By doing so he actually rejects the very notion of Actual Infinity in terms of Platonic Infinity (which according to it there is a complete set of infinitely many things).

By rejecting Actual Infinity in terms of Platonic Infinity, it has to be understood that ZF(C) (which includes the Axiom Of Infinity) can't be used in order to establish even the infinite set of all natural numbers.

 

[jsfisher]

Ok, enough.

jsfisher wishes to add more objects to A, which is equivalent to the infinite set of all natural numbers.

So, you want A to be the set of natural numbers. Why didn't you just say so in the first place? For that matter, why not use the more conventional letter N as the label for the set?

Ok, now what? How are you proposing to transition over to Godel's Incompleteness Theorems?

 

[doronshadmi]

Huh? The Axiom of Infinity at no point says that "no member is missing"

In that case no complete infinite set is established by it ("By by Cantorean Actual Infinity" , "Welcome Potential Infinity").

So, you want A to be the set of natural numbers. Why didn't you just say so in the first place? For that matter, why not use the more conventional letter N as the label for the set?

Ok, now what? How are you proposing to transition over to Godel's Incompleteness Theorems?

In case that you are still missing it, all it matters is that Infinitely many things are taken in terms of Platonic Infinity, which means that such collection is complete and therefore inconsistent by GIT.

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

No one of A members, which are encoded by Godel numbers (which are actually the set of all natural numbers) is missing.

So A is complete and therefore inconsistent by GIT.

Moreover, the difference between cardinal and ordinal numbers has no impact on the discussed subject in case that Infinity is taken in terms of Platonic Infinity (as already given in http://www.internationalskeptics.com/forums/showpost.php?p=12777063&postcount=3322).

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For those who have missed it, Gödel was a Platonist (he agreed with Actual infinity in terms of Cantor) and his main motivation behind his Incompleteness Theorems was to logically demonstrate that formal systems that are strong enough in order to deal with Arithmetic, can't be complete AND consistent and also can't prove their own consistency (which means that many "interesting" formal systems can't deal with Platonic realms).

But Gödel's Incompleteness Theorems also prove that the very notion of Actual infinity in terms of Platonism (which is also Actual infinity in terms of Cantor) does not hold logically (at least in the strong sense).