After the failure of Hilbert’s original program due to Gödel’s second incompleteness theorem, relativized Hilbert’s programs have been sug-gested. While most metamathematical investigations are focused on car-rying out mathematical reductions, we claim that in order to give a full substitute for Hilbert’s program, one should not stop with purely mathe-matical 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 nec-essarily 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 no-tation 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. 1 Reducing Theories to Trustworthy Principles In the early 1920’s Hilbert suggested a program for the foundation of mathemat-ics, which is now called Hilbert’s program. As formulated in [40], “it calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof it-self was to be carried out using only what Hilbert called ’finitary ’ methods. The special epistemological character of finitary reasoning then yields the required justification of classical mathematics. ” Because of Gödel’s second incomplete-ness theorem, Hilbert’s program can be carried out only for very weak theories
