Abstract

Gödel’s two incompleteness theorems are usually grouped together as a single result, but conceptually they are very different. In particular, I will talk about several “counterexamples” of Gödel’s second incompleteness theorem, which says that a computable theory containing enough arithmetic cannot prove its own consistency. The existence of such examples highlights the question of what the correct conceptual understanding of the statement of second incompleteness theorem should be. I will offer a solution based on categorical logic, where we can identify universal property of arithmetic. The incompleteness phenomena will then be consequences thereof.