Do Gödel's First Incompleteness Theorem imply the inconsistency of Platonic Infinity?

Question

According to Modern Mathematics (where the majority of mathematicians agree about the notion of actual infinite sets, as established mostly by George Cantor) an inductive set (as given by ZF(C) Axiom Of Infinity) has an accurate cardinality, which implies that it is complete (no one of its members is missing).

In other words, by ZF(C) Axiom Of Infinity there exists at least one infinite AND complete set (if we agree with the notion of actual infinity, as mostly established by Cantor).

Now, assume a complete set of infinite axioms (according to the reasoning of actual infinity, as established mostly by Cantor and agreed by the majority of modern mathematicians).

But by Gödel's First Incompleteness Theorem such set of axioms must be inconsistent as follows:

Set A (which is strong enough is order to deal with Arithmetic) is a set of infinitely many axioms (where each axiom is written by finitely many symbols) which is established by using ZF(C) Axiom Of Infinity on ZF(C) itself, such that Infinity is taken in terms of Platonic Infinity (By Platonic Infinity there exists a set of infinitely many things as a complete whole (without using any process)).

Some example: The infinite set of all natural numbers is taken in terms of Platonic infinity.

Now all we care is about the set of all infinitely many wffs (in terms of Platonic Infinity) that are established in A .

Each wff has some Gödel number, where at least one of these wffs, called G, states "There is no number m such that m is the Gödel number of a proof in A , of G" (since G needs a proof, it is not an axiom but a theorem).

Since all wffs are already in A and all Gödel numbers are already in A (because Infinity is taken in terms of Platonic Infinity) there is a Gödel number of a proof of G in A , which contradicts G in A , exactly because A is complete (as shown) and therefore inconsistent.

So the problem is actually the notion of a complete set of infinity many things in terms of Platonic Infinity, and in order to save the consistency of A, ZF(C) Axiom Of Infinity is taken in terms of Potential Infinity (process is used, exactly as done in case of GIT in its standard sense).

But then ZF(C) Axiom Of Infinity can't be used in order to establish sets in terms of Platonic Infinity (for example: the notion of The infinite set of all natural number is logically inconsistent).

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Gödel was a Platonist (he agreed with Actual infinity in terms of Cantor (which is actually Platonic Infinity)) 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 First Incompleteness Theorem also proves that the very notion of Actual infinity in terms of Platonism (which is also Actual infinity in terms of Cantor) does not hold logically.

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There is a non-interesting solution about the discussed subject, as follows:

G states: "There is no number m such that m is the Godel number of a proof in A , of G"

If G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Platonic Infinity) it is actually a wff that is true in A , which does not have any Gödel number that is used in order to encode G's proof (since axioms are true wff that do not need any proof in A ).

But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).

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Also please be aware of the following:

1) If ZF(C) Axiom Of Infinity is not necessarily taken in terms of Platonic Infinity, then ZF(C) Axiom Of Infinity is taken in terms of Platonic Infinity OR Not (useless tautology).

2) If ZF(C) Axiom Of Infinity is not necessarily taken in terms of Platonic Infinity, then it can't be used in order to establish even the set of all natural numbers (which means that N (and | N |) is not necessarily established by ZF(C)).

Answer 1

Throughout, I'm assuming that ZFC is consistent.

There are a lot of confusing points here, and I don't really understand what you're setting up with bijections. However, I believe the key mistake you make isn't actually related to sizes of sets at all, but rather a serious misapplication of the incompleteness theorem. The language about infinite sets and bijections simply serves to obfuscate this issue.

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Ultimately you're building an extension of ZFC gotten by repeatedly adding Godel-type sentences until we've "gotten all of them." You then claim that this theory is complete and consistent as long as ZFC is and that GIT applies to it, yielding a contradiction. However, each of these claims is flawed.

Let's start with a minor observation: that there's no reason for the resulting theory to be complete: there are sentences independent of ZFC which are not "Godel-type," and indeed are not implied by any consistent "iterated Godel sentences" - the continuum hypothesis is one of these. The reason this is a minor observation is that all you really need is a consistent recursively axiomatizable extension of ZFC which proves its own Godel sentence.

However, it turns out that this is fatally flawed too. In trying to whip up such an extension, you need some process for adding axioms which decide every sentence (or every Godel sentence). But no matter how you do this, the resulting theory won't be recursively axiomatizable since telling whether a sentence is independent of ZFC is undecidable. And GIT does not apply to non-recursively-axiomatizable theories (and trying to make it recursive by "guessing" at independence in a recursive way will result in an inconsistent theory).

And along the same lines, there are subtleties to iterating consistency principles. These are a bit technical to get into, but essentially when you actually sit down to make everything precise you're essentially going to wind up "iterating the Godel construction along computable well-orderings" - but the set of these is far from recursive. And if you try to "overshoot" to get enough axioms but still stay recursive, it turns out that you wind up with an inconsistent theory.

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You should read about iterated consistency principles - understanding these will clarify the issues here. The topic is treated in several questions at math.stackexchange and at Mathoverflow, and also in several papers and books (and I recommend Franzen's book Inexhaustibility in particular).

Answer 2

ZF with the axiom of infinity removed is not complete. It can be modeled by the collection of finite sets, and it has a model with an infinite set. Since these two models are not isomorphic, ZF without the axiom of infinity is incomplete. Presumably the model consisting of the collection of finite sets establishes that ZF without the axiom of infinity is consistent.

Gödel Incompleteness says that ZF (which includes the axiom of infinity) is either inconsistent or incomplete.

Answer 3

Completeness has to do with the derivation system. Can that derivation system derive for all statements either the statement or the negation of the statement? If it can then it is complete.

Regarding the first incompleteness theorem, the following comes from the Wikipedia source referenced by the OP:

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.

If the derivation system for the arithmetic of the natural numbers contained a set of axioms, infinite in quantity or not, that allowed the derivation system to prove all semantically valid (true) statements, then those axioms would be inconsistent.

It may be useful to put this in perspective.

Suppose a set of axioms of propositional logic is inconsistent. Then from those axioms one could derive both P and ~P. If one could derive both of these contradictory statements then any well-formed formula Q of propositional logic would be semantically valid based on the following truth table:

It would also be syntactically valid. One could also derive any Q given both P and ~P as premises using the explosion (X) inference rule:

Any formal system that could do that is complete, but in an undesirable way. Because every formula can be derived, and since the derivation system is sound, every formula is true, the system is trivial. That is something one wants to avoid.

So one wants to negate the following:

inconsistent and complete

Using De Morgan's laws the negation of the above is:

consistent or incomplete

The or allows one to have systems that are both consistent and complete, as long as the consistent case remains true, such as propositional logic. Godel showed, using the arithmetic of the natural numbers, that one should not expect all formal systems to have both.

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Here is the question:

In this case, isn't ZF(C) Axiom Of Infinity is some kind of a "Trojan horse" that prevents the consistency of ZF(C)?

Having an infinite number of axioms does not make a set of axioms inconsistent. Consistent axiom schemas allow for a countably infinite number of axioms by substituting formulas in schematic variables. Rather inconsistency means being able to derive both a proposition and its negation from the axioms.

Godel's incompleteness theorem shows that there exists derivation systems, such as those for the arithmetic of natural numbers, that cannot be both consistent and complete. Such systems can still be consistent and incomplete.

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Michael Rieppel. Truth Table Generator. https://mrieppel.net/prog/truthtable.html

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

Wikipedia contributors. (2019, July 29). Gödel's incompleteness theorems. In Wikipedia, The Free Encyclopedia. Retrieved 12:31, August 4, 2019, from https://en.wikipedia.org/w/index.php?title=G%C3%B6del%27s_incompleteness_theorems&oldid=908327466