From the conclusion of “Cantor’s Diagonalization Method” by Alexander Kharazishvili (and I’ll assume the reader has some familiarity with Gödel’s famous theorems):
Tarski’s theorem leads to Gödel’s first incompleteness theorem. Let Pr denote the set of natural numbers corresponding to the provable sentences in formal arithmetic. Pr is recursively enumerable. Since any recursively enumerable set of integers is arithmetical, Pr cannot coincide with Tr [“the set of all natural numbers that correspond to those sentences true in the standard model of formal arithmetic”]. It can only be a proper subset of Tr. Truth and provability are not the same. That this discovery was revolutionary hardly does justice to its significance. Thus, there is a sentence S that is true, but not provable, in the standard model of formal arithmetic. Neither S, nor its negation, ¬S, is provable and formal arithmetic is incomplete. A similar argument is applicable to any recursive mathematical theory that is stronger than formal arithmetic (e.g., ZFC set theory).
We’ve talked about this many times: the idea that reason and rationality are enough, or are the ultimate (as in best) form of thought, are obviously false ideas. They are useful and necessary, but alone they are insufficient.
Reason, in the way I am thinking about it, is almost mechanistic. It takes a pile of thoughts, processes them by known rules, and spits out ideas. The processing is like a machine, which can be souped up or in poor repair. However complex it is, it isn’t anything more than brute mental force.
The ore at the base of the pile of thoughts must be supplied from outside. Reason needs fuel it can’t provide itself. Reason operating only itself is like a perpetual motion machine, an impossibility.
It is not reason that concludes Reason is reasonable. It is not Reason that proves truths that cannot be proved by Reason. “Truth and provability are not the same.” Provability by Reason, that is. Proof by other mechanisms is still possible; indeed, necessary.
All mathematics proceeds from two things: (1) unproveable-by-Reason propositions (axioms) and (2) unproveable-by-Reason belief Reason will work to generate new true propositions. This sequence does not apply to mathematics alone, but all thought.
Therefore there must be something beyond Reason providing our deepest and most consequential truths.