On the Nature of Gödel’s Second Incompleteness Theorem

In several recent articles, we have shown how such totality conditions are related to both generalizations and boundary-case exceptions for the Second Incompleteness Theorem. For instance, our article [2] showed how essentially all axiom systems recognizing addition and multiplication as “total” functions are unable to prove a theorem affirming their semantic tableaux consistency. In contrast, [1] showed exceptions to the semantic tableaux version of the Second Incompleteness Theorem are feasible when an axiom system recognizes addition but not multiplications as a total function. In this talk, we will outline several more recent results we have published during the last 2 years: