Abstract. On the face of it, Hilbert’s Program was concerned with proving consistency of mathematical systems in a finitary way. This was to be accomplished by showing that that these systems are conservative over finitistically interpretable and obviously sound quantifier-free subsystems. One proposed method of giving such proofs is Hilbert’s epsilonsubstitution method. There was, however, a second approach which was not refelected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert’s first epsilon theorem and a certain “general consistency result. ” An analysis of this so-called “failed proof ” lends further support to an interpretation of Hilbert according to which he was expressly concerned with conservatitvity proofs, even though his publications only mention consistency as the main question. §1. Introduction. The aim of Hilbert’s program for consistency proofs in the 1920s is well known: to formalize mathematics, and to give finitistic consistency proofs of these systems and thus to put mathematics on a “secure foundation.” What is perhaps less well known is exactly how Hilbert thought this should be carried out. Over ten years before Gentzen developed sequent calculus formalizations

Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s

Hilbert’s ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.

We introduce a nonstandard arithmetic NQA − based on the theory developed by R. Chuaqui and P. Suppes in [2] (we will denote it by NQA +), with a weakened external open minimization schema. A finitary consistency proof for NQA − formalizable in PRA is presented. We also show interesting facts about the strength of the theories NQA − and NQA +; NQA − is mutually interpretable with I∆0 + EXP, and on the other hand, NQA + interprets the theories IΣ1 and WKL0.

Abstract. Hilbert’s ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure. Keywords: Hilbert’s ε-calculus, epsilon theorems, Herbrand’s theorem, proof complexity 1.