The Use of Trustworthy Principles in a Revised Hilbert’s Program

After the failure of Hilbert’s original program due to Gödel’s second incompleteness theorem, relativized Hilbert’s programs have been suggested. While most metamathematical investigations are focused on carrying out mathematical reductions, we claim that in order to give a full substitute for Hilbert’s program, one should not stop with purely mathematical investigations, but give an answer to the question why one should believe that all theorems proved in certain mathematical theories are valid. We suggest that, while it is not possible to obtain absolute certainty, it is possible to develop trustworthy core principles using which one can prove the correctness of mathematical theories. Trust can be established by both providing a direct validation of such principles, which is necessarily non-mathematical and philosophical in nature, and at the same time testing those principles using metamathematical investigations. We investigate three approaches for trustworthy principles, namely ordinal notation systems built from below, Martin-Löf type theory, and Feferman’s system of explicit mathematics. We will review what is known about the strength up to which direct validation can be provided.