Escaping Gödel's proof

Question

Is there any way in which a reasonably strong foundation of mathematics can get around the hypotheses of the incompleteness theorems?

Answer 1

Essentially, the answer is no (in classical first-order logic). If your theory,T, is capable of "understanding" both addition and multiplication of $\mathbb{N}$, then one of the following fails: (1) $T$ is consistent, (2) $T$ is recursive, (3) $T$ is complete.

Giving up (1) is the absolute worst case for a logician/mathematician. The only time someone would think about giving up this property is if they thought that the theory of the natural numbers is inconsistent. Essentially, this position might be held by an ultrafinitist or someone who believe PA is inconsistent (see Edward Nelson).

Giving up (2) is useless for the logician/mathematician. There does exist a complete and consistent theory of arithmetic, namely $Th(\mathbb{N})$ in the language $L=\{+,\times,0,1\}$. However, the theorems of $Th(\mathbb{N})$ cannot be found recursively, i.e., there is no algorithm that tells us whether or not a sentence in the language of arithmetic is a theorem of $Th(\mathbb{N})$. Therefore, $Th(\mathbb{N})$ is boring to study.

Giving up (3) is the most reasonable for research purposes and for "real-world" study. There are still open problems in basic arithmetic (see Goldbach Conjecture) and using PA to try and solve this problems is still possible.

(As an asside: Note furthermore, that if GC is independent of PA, we can show, using ZFC, that $Th(\mathbb{N})\vdash GC$).