Are (some) axioms "unprovable truths" of Godel's Incompleteness Theorem?

Question

Like any math newbie, Godel's Incompleteness Theorems are easy to understand in general layman's terms, but difficult to understand beyond the typical "liar's paradox" and "barber's paradox" type examples.

But then I started thinking, are axioms examples of the truths of mathematics that can't be proven? For example, Peano's Postulates: a very popular "starting point" for deducing other mathematical truths. One of the postulates is, "0 is a natural number." Well wait a second: what is "0"? What is a "natural number"? In order to take that axiom as a truth, there must be a definition for what 0 is, and what a natural number is. But even if we assign them definitions in English, can those definitions be proven?

If I follow this train of thought, I eventually find myself completely outside of mathematics, and more into philosophy --- can we even prove what a "natural number" actually is? If I'm not careful, eventually I end up in lala land thinking about the meaning of existence and reality itself.

Does this even make sense? My mind has been blown so many times on this topic, I spend more time scooping my brain off the floor than forming coherent questions.

Answer 1

To the extent that our "axioms" are attempting to describe something real, yes, axioms are (usually) independent, so you can't prove one from the others. If you consider them "true," then they are true but unprovable if you remove the axiom from the system. In that sense, the smaller system has "true" but unprovable theorems.

But the "trueness" of Gödel's statement is a bit more complicated.

Let's say it turned out that the Goldbach conjecture was undecidable. To me, that would mean that it is "intuitively true," since if it was false, we could find a counter-example that was a finite statement. The fact that we can't provide a counter-example, however, is not enough to prove it is true. This might seem strange, even absurd.

One way I like to think of it is that proofs are finite things, but we are often trying to prove things about infinitely many numbers. Induction, for example, can be thought of as a finite way of outlining an infinite proof.

Intuitively, what Gödel showed is that (under enough complexity) there are always theorems that have infinite proofs but for which there are no finite proofs. For example, Goldbach might be one of those cases. Each $2n, n\geq 3$ might be expressible as the sum of two primes, but if we can't outline that proof finitely, we are stuck.

A statement like Gödel's - rougly, "This theorem does not have a (finite) proof from these axioms" - is such an example. We can enumerate all finite logical proofs, and check if it proves our theorem. This would yield an infinite proof of the result. So in that sense, it is "true." But it obviously can't be proven.

The fact that we can't resolve this statement from these axioms means, intuitively, that it is true. Our intuition about the natural numbers says this ought to be true, and that our axioms didn't fully capture our intuition.

Answer 2

Gödel's incompleteness theorem has more to say about decidability than mathematical Platonism. It's not correct to say there is anything that is "true" or "false" just whether axioms are independent, theorems, or inconsistent.

There are a number of axioms in orthadox set theory that are provably independant; The Axiom of Choice, the Diamond Principle, the Continuum Hypothesis all come to mind. But its incorrect to say that these are true or false outside the philosophy of mathematics, because assuming the negation of these axioms yields consistent theories as well.

When you say that the incompleteness theorems imply there are unprovable but true statements, we mean true in some larger system that the smaller system (in the first incompleteness theorem, Peano Arithmetic) is encapsulated in.

An example of an "unprovable but true" statement in PA is Goodstein's Theorem, in that it can be proven in ZFC but can't in Peano arithmetic.

We can reasonably assume that there are undecidable statements in ZFC where their undecidability implies their truth value.

In mathematical formalism, "truth" is a value that has a precise definition that is tangentially related to the metaphysical meaning we associate with when discussing philosophy, and so what Gödel's incompleteness theorem says about the Axioms isn't much more than they are possibly undecidable with respect to other axioms.

Answer 3

Axioms are assumed to be true, they can (if the axioms are independent in the axiomset) not be proved.

For any type of fundamental reasoning you need to start with some assumptions and those assumptions are the axioms and the rules of inference.

The big advantage of axiomatic reasoning (using axiom(scheme)s and modus ponens) is that no other assumptions can "sneak into" the reasoning.
(for example natural deduction presupposes P->(Q -> P) , (P-> (Q->R)) -> (Q ->(P -> R)) and (P->(P->Q)) -> (P->Q), and these can be proved. ex nihilo, from nothing at all)

You can have a look at the axioms and think that doesn't look true but removing or replacing axioms can lead to a different logical system.

PS1 this is not always the case logics are defined by their set of theorems not by their axioms.
so if all theorems of logic 1 are theorems of logic 2 and all theorems of logic 2 are theorems of logic 1 then logic 1 and logic 2 are the same logic.

An example where replacing axioms does lead to different logics: Some axiomsets of Classical logic contain as axiom:
(~p -> ~Q) -> (Q -> P)

Intuitionistic and constructive logicians disagree with this theorem (it implies the law of excluded middle) and replace this (~p -> ~Q) -> (Q -> P) with the axioms:

(P-> ~Q) -> ((P->Q) -> ~P)
and
P -> (~P -> Q)

Minimalist logicians disagree with the P -> (~P -> Q) (it is the law of ex-falso-quidlibed , of contradiction everything follows) and removed this axiom so only keeping:
(P-> ~Q) -> ((P -> Q) -> ~P)

Sub-minimalist logicians disagree with (P-> ~Q) -> ((P -> Q) -> ~P) and maintain that negation is fully defined by:
(P -> Q) -> (~Q -> ~P) (this is a theorem in classical, intuitionistic and minimal logic)

In each of these logics the set of theorems differ and therefore they are different logics.

Answer 4

Axioms are not 'statements unprovable by Godel', but 'statements taken to be true'. If you take a unprovable statement, and start using it as 'true', then it becomes an axiom.

For example, there is in Geometry, the so-called 'fifth postulate', or parallel axiom. There is a certian amount of theory that can be developed without it. The fifth postulate has many expressions, such as the sum of a triangle being two right angles.

Since it is not provable, one can then consider the outcomes when it assumes separate values. When the triangle angles sum to more than two right angles, the postulate that lines can be extended as far as one wants becomes invalid. But this geometry becomes the spherical geometry. When the angles of a triangle sum to less than two right-angles, the geometry proceeds, producing different, but consistant results (of hyperbolic geometry).

So, in place, one can think of the incompleteness theory as proto-axioms, but they do not become axioms as such unless you start making further theory depend on it.

Answer 5

Yes, this is philosophy. You do not understand mathematics by understanding some other mathematics, somewhere this should stop.

All this reduces to rule following, which is an act of a human. What is 0? You learn it when you learn counting. 0 is the thing you add and nothing changes. Mathematics is embedded into natural language and our life form. You do not learn counting by Peano axioms, this comes much later. You learn counting by counting apples, a language game in the phyisical world.

This is very much a Wittgensteinian way of thinking. Read his work, unfortunately I cannot give you precise references but it might be the the Blue and Brown books already contains some of this, and for sure the Remarks on the foundation of mathematics will lead you on this path.