Is all mathematical truth simply the product of formal calculations, more precisely the result of formal proofs from formal axioms? This is arguably the most basic question in the Philosophy of Mathematics.

The incompleteness theorems of Gödel argue that the answer is no. But the extent to which this is actually a compelling answer is debatable.

However, there are new proofs of the incompleteness theorems which are based on computational complexity, and which definitively show that mathematical truth transcends mathematical proof.

Nowhere is the issue of truth versus proof more central than in Set Theory, which is the mathematical study of Infinity. Here many of the most basic questions, such as that of Cantor’s Continuum Hypothesis (CH), are known to be beyond the reach of proofs from the accepted ZFC axioms of Set Theory. So any resolution of the problem of CH must involve truth beyond proof.