The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program

Abstract

Abstract. After a brief flirtation with logicism in 1917-1920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Göttingen between 1917 and 1923, and notably in Ackermann's dissertation of 1924. The main innovation was the invention of the ε-calculus, on which Hilbert's axiom systems were based, and the development of the ε-substitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the ε-notation and provides an analysis of Ackermann's consistency proofs for primitive recursive arithmetic and for the first comprehensive mathematical system, the latter using the substitution method. It is striking that these proofs use transfinite induction not dissimilar to that used in Gentzen's later consistency proof as well as non-primitive recursive definitions, and that these methods were accepted as finitistic at the time.

The Epsilon Calculus and Herbrand Complexity

“…The ε-calculus was originally developed in the context of Hilbert's program of consistency proofs. Early work in proof theory (before Gentzen) concentrated on the ε-calculus and the ε-substitution method and was carried out by Ackermann [1,2] (see also [19]), von Neumann [24], and Bernays [14] (see also [25,26]). The ε-calculus is of independent and lasting interest, however, and a study from a computational and proof-theoretic point of view is particularly worthwhile.…”

Tarski’s Practice and Philosophy: Between Formalism and Pragmatism

“…42 strikingly similar to Weyl's viewpoint in Weyl [79]. On his side, Zach [86] establishes (p. 220) that the general schema of primitive recursion was already mentioned in Hilbert's unpublished course of 1921-1923. Moreover, he argues that Hilbert's outlook was markedly different from Skolem's [55] (suggesting that there was no influence either way).…”

“…I would like only to point out some philosophical modifications it involved. 29 Hilbert supported indeed 28 See in particular the recent paper by R. Zach [86] 30 and to play fully the formula game in a constructive way. a) Hilbert was urged by Brouwer's and Weyl's objections to make precise the concept of formal system through a kind of material implementation.…”

Gödel on intuition and on Hilbert's finitism

“…(We will discuss the connection between P RA and the system A presently.) Literally, Gödel is mistaken about this: As Richard Zach has noted [Zach, 2003], Hilbert had approved as finitist Ackermann's use of induction up to ω ω ω in his dissertation [Ackermann, 1924]. In fact, though, the only instance of transfinite induction for which Ackermann gave any justification in his paper was induction up to ω 2 and his argument was essentially just the reduction of this to primitive recursion with substitution in the parameters (which of course reduces to primitive recursion).…”

“…In particular, although the issue has been debated on the battlefield of §7 of Grundlagen der Mathematik, Volume 1 [Hilbert and Bernays, 1934] (see [Zach, 1998], [Tait, 2002] and [Tait, 2005b, Appendix]), the question of whether Bernays, in the period 1922 until 1931, when the quest for consistency proofs was framed in terms of finitism and before the need to extend that framework became apparent, regarded reducibility to P RA as the essential mark of finitist definition and proof is not, in my opinion, definitively resolved. On the other hand, the absence of a clear statement of that criterion could lead one to the presumption that they did not accept it.…”