Hilbert's Program and Infinity

TL;DR

Hilbert's program seeks a consistency proof for classical mathematics using finitary means by eliminating the infinite via the epsilon-calculus. The paper explains how epsilon terms $\varepsilon_x A(x)$ replace quantifiers and analyzes the epsilon-substitution method, which transforms ideal derivations into finitary, real derivations and aims for conservativity over a finitary subsystem. It provides a detailed account of the method's handling of epsilon-types and the solving substitutions needed to remove epsilon-terms, then discusses the metatheoretical induction on ordinal weights (up to $\varepsilon_0$) required to justify termination. The work highlights that while such transformations remove explicit infinitary content, the justification of termination itself relies on infinitary reasoning about ordinals, underscoring enduring philosophical questions about the foundations of mathematics and the reach of finitary proofs.

Abstract

The primary aim of Hilbert's proof theory was to establish the consistency of classical mathematics using finitary means only. Hilbert's strategy for doing this was to eliminate the infinite (in the form of unbounded quantifiers) from formalized proofs using the so-called epsilon substitution method. The result is a formal proof which does not mention or appeal to infinite objects or "concept-formations." However, as later developments showed, the consistency proof itself lets the infinite back into proof theory, through a back door, so to speak. The paper outlines the epsilon substitution method as an example of how proof-theoretic constructions "eliminate the infinite" from formal proofs, and how they aim to establish conservativity and consistency. The proof also requires an argument that this proof theoretic construction always works. This second argument, however, requires possibly infinitary reasoning at the meta-level, using induction on ordinal notations.