How can you come to the truth of a statement without proving it?

Question

I was reading a bit about Gödel's incompleteness theorems. I haven't took the time to really study it, but I'm very curious about statements like these:

In other words, if our axioms are consistent then in every model of the axioms there is a statement which is true but not provable. source

And

Given any system of axioms that produces no paradoxes, there exist statements about numbers which are true, but which cannot be proved using the given axioms.

What I don't understand is this. How can you show that such a statement is true, without proving it ? This seems like a contradiction in itself to me.

Can someone give me an example of such a statement (about numbers) that we know is true, but which cannot be proven to be true ? And how then do you conclude that such a statement is true ? Because of the relations that numbers have with the real world ?

Answer 1

True and false are relative to a structure, these are semantics properties of a sentence in a given interpretation of the language.

In some cases, like in the case of arithmetics, when we say that a statement is true we mean that it is true in a very specific model. In the case of arithmetical theories (like $\sf PA$ for example) we take the model to be $\Bbb N$.

So to determine if a statement about the natural numbers is "true" we need to see if it is true in $\Bbb N$.

Provable, again, depends on the theory. The axiom of choice is not provable from $\sf ZF$ but it is most certainly provable from $\sf ZFC$. So when we just say that something is provable or unprovable we need to have a proper context to give a correct interpretation of the statement.

In the case of $\Bbb N$ and the natural numbers, this is commonly Peano axioms, $\sf PA$.

So when we say that the statement "Every Goodstein sequence terminates" is true but unprovable, we really say that it is true that in $\Bbb N$ every Goodstein sequence terminates, but it is not true in every model of $\sf PA$.

Other true, but unprovable statements may include various consistency claims (e.g. $\operatorname{Con}\sf (PA)$ is true but unprovable) and other similar results from related incompleteness proofs. And we can encode many of them in the form of Diophantine equations or polynomials (see this particular example).

(All this, of course, has nothing to do with the real world.)

Answer 2

In the First Incompleteness Theorem, the word 'proof' does not refer to the general informal style of argument that people create to establish results. It instead has the far more specific meaning 'a derivation from axioms in formal system $X$'. The theorem's proof (in the informal sense) shows how to construct a sentence $S$ (i.e a first-order formula with no free variables) in the formal system $X$, which we humans interpret as (metaphorically) 'saying' "Sentence $S$ has no derivation in formal system $X$". *From outside of formal system $X^*$, we humans can see that if $S$ actually DID have a derivation, $S$ would be false. If system $X$ is consistent (and not perverse), as we expect, then only true sentences should be derivable. So $S$ must not be derivable.

So $S$ is 'true but not provable' - meaning that $S$ is not provable - meaning not derivable in system $X$ - but we have proved --from outside of system $X$-- that it is true.