[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.