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

  title={The Use of Trustworthy Principles in a Revised Hilbert’s Program},
  author={Anton Setzer},
After the failure of Hilbert’s original program due to Godel’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… 
2 Citations

Gentzen’s Consistency Proof in Context

Gentzen’s celebrated consistency proof—or proofs, to distinguish the different variations he gave1—of Peano Arithmetic in terms of transfinite induction up to the ordinal2 \(\varepsilon _{0}\) can be

An Upper Bound for the Proof-Theoretic Strength of Martin-Löf Type Theory with W-type and One Universe

We present an upper bound for the proof theoretic strength of Martin-Löf’s type theory with W-type and one universe. This proof, together with the well ordering proof carried out in [Set98b] shows



Hilbert's program then and now

Hilbert's program relativized; Proof-theoretical and foundational reductions

Here a body of proof-theoretical results stemming from H.P. are surveyed in a way that is closely tied to various reductive foundational aims, albeit going beyond those advanced by Hilbert.


  • G. Kreisel
  • Mathematics
    The British Journal for the Philosophy of Science
  • 1953
IN Hilbert's theory of the foundations of any given branch of mathematics the main problem is to establish the consistency (of a suitable formalisation) of this branch. Since the (intuitionist)

A Survey of Proof Theory

One might fairly say that the very meaning of our subject has changed since Hilbert introduced it under the name Beweistheorie (it was meant to be the principal tool for formulating Hubert's general

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This work bears on the question of the indispensability of set-theoretic foundations for mathematics, and aims to give a constructive and predicative consistency proof for a classical theory T in which large parts of infinitistic mathematics can be developed.


Hilbert's plan for understanding the concept of infinity required the elimination of non-finitist machinery from proofs of finitist assertions. The failure of the original plan leads to a hierarchy

Does Reductive Proof Theory Have A Viable Rationale?

The goals of reduction andreductionism in the natural sciences are mainly explanatoryin character, while those inmathematics are primarily foundational.In contrast to global reductionistprograms

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  • P. Dybjer
  • Philosophy
    Epistemology versus Ontology
  • 2012
It is proposed that testing for impredicative type theory should be based on the evaluation of open expressions, in contrast to the testing semantics for Martin-Lof’s predicative intuitionistic type theory which is based onThe evaluation of closed expressions.

Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics

  • S. Feferman
  • Philosophy
    PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association
  • 1992
Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover,

An extended predicative definition of the Mahlo universe

In this article we develop a Mahlo universe in Explicit Mathematics using extended predicative methods. Our approach differs from the usual construction in type theory, where the Mahlo universe has a