On the Results of a 14-Year Effort to Generalize Gödel’s Second Incompleteness Theorem and Explore Its Partial Exceptions

The Second Incompleteness Theorem states that axiom systems of sufficient strength are unable to verify their own consistency. Part of the reason that Gödel's theorem has fascinated logicians is that it almost defies common sense. This is because when human beings cogitate, they implicitly presume that thinking is a useful process. However, this tacit assumption would initially appear to presume that logic is consistent (since an inconsistent logic formalism is known to prove all true and false theorems — thereby rendering logical deduction quite useless). Thus, the Second Incompleteness Theorem seems to suggest that it is almost impossible — and certainly very awkward — to formalize the common-sense assumption that thinking is useful (because sufficiently strong axiom systems are unable to recognize their own consistency.) The preceding mystery about the nature of incompleteness lies at the heart of the scholarly community's fascination with Gödel's historic discovery [3]. Several of Gödel's biographers [4, 21] have also noted that Gödel explicitly hedged in one of the closing paragraphs in his paper [5] about whether or not the Second Incompleteness Theorem should be interpreted as meaning all efforts for an axiom system to verify its own consistency by finite means are baseless. For instance, page 58 of Yourgrau's biography [21] of Gödel points out that Von Neumann viewed the Second Incompleteness Theorem as having a much broader range of applications than Gödel for several years after the publication of Gödel's seminal 1931 paper. What we have sought to do during the last 14 years was to simultaneously explore paradigms where an unusual axiom system can formalize at least a partial conception