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

  • Anton Setzer


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.


Type Theory Proof Theory Elimination Rule Introduction Rule Incompleteness Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The author wants to thank the anonymous referees for extraordinarily detailed refereeing and many very valuable comments; Fredrik Nordvall Forsberg and Håkon Gylterud for careful proof reading; and Reinhard Kahle for his encouragement to writing such a rather philosophical article and for his patience while waiting for the completion of this article. Research for this article was supported by EPSRC grant EP/G033374/1.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceSwansea UniversitySwanseaUK

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