Hilbert's Program and Infinity

TL;DR

This paper discusses Hilbert's proof theory aiming to establish the consistency of classical mathematics through finitary methods, focusing on the epsilon substitution method and its limitations involving infinitary reasoning.

Contribution

It analyzes the epsilon substitution method as a proof-theoretic technique to eliminate the infinite from formal proofs and discusses its limitations involving infinitary meta-level reasoning.

Findings

The epsilon substitution method aims to eliminate the infinite from proofs.

Consistency proofs can reintroduce the infinite via meta-level reasoning.

Infinitary induction is necessary for the method's validity.

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.