[zajio1am]

> must necessarily contain true (resp. false) facts for which no (resp. counter) proof can be given

This formulation misses the important aspect that whether the statement is 'true' is not absolute property (outside logical truths). We can consider truthfulness of a statement in a specific structure or in a specific theory.

E.g. a statement can be undecidable in Peano arithmetic (a theory) while true in natural numbers (a structure, model of Peano arithmetic), but that just means there is a different structure, different model of Peano arithmetic in which this statement is false.

[Tainnor]

Discussing Gödel in terms of "truth" immediately brings about deep philosophical discussions about what that actually means in the context of mathematics. If the "true" nature of the (standard model of) arithmetic is unknowable in full, does it even exist?

I like to be pragmatically classical in my mathematical outlook (I don't worry about LEM), but when we come to foundations, I find that we need to be a little bit more careful.

Gödel's original proof (or rather, the slightly modified Gödel-Rosser proof) avoids notions of absolute truth - it says that for any consistent model of arithmetic, there must exist undecidable sentences. These are ultimately purely syntactical considerations.

(I'm not claiming that there is no relation to "truth" at all in this statement, just that things become less precise and more philosophical once that's how we're going to frame things)

[zajio1am]

These deep philosophical discussions are about absolute "truth" in general, but in logic, the truthness of formula in specific mathematical structure is well-defined through structural induction over the formula, although with non-finite, non-computable steps. Yes, from strict-finitistic point of view even the concept of truth in the standard model of arithmetic could be problematic.

[Tainnor]

I'm personally not a constructivist or even finitist (mostly for pragmatic reasons), but discussions about Gödel get sidetracked by musings about either of these philosophies so often that I feel it's important to point out that there is an understanding of Gödel's theorems that can be understood and discussed regardless of one's commitment to a particular philosophy of mathematics (well, at least assuming some "reasonable" base).

If you are a finitist, you could interpret Gödel's theorem to mean that infinite systems such as the natural numbers don't actually exist, because they can never be understood in full (I'm not a finitist, so maybe I'm wrong in this conclusion). If you're a classical platonist, they would mean that the "natural numbers" do exist in some platonic realm, but will never fully be captured by human mathematics. If you're a formalist like me, maybe you'd say that it's very useful to pretend that the natural numbers exist, but that strictly speaking we don't really fully understand what they are or should be (but it turns out not to matter all that much).

Either way, a complete theory of the natural numbers doesn't exist.