On the power of proof: characterizing the gap between provability and truth

My thesis seeks to distinguish between mathematical terms that provide a strong characterization of correctness; namely, proof and truth. We first formalize our motivating question, the Henkin Problem, and subsequently answer it by proof of Löb’s Theorem. Gödel’s 2nd Incompleteness Theorem, a corollary to Löb’s result, inspires a perspective on meta-mathematics that employs modal logic through the framework of Kripke models. As such, we first model PA-provability and then move towards modeling ZFC-provability via the use of forcing. We ultimately conclude that mathematical uncertainty stems from the inaccessibility of “natural” truth, which must be resolved by use of intuition.