The Collapse of the Hilbert Program: A Variation on the G\"odelian Theme

TL;DR

This paper demonstrates a fundamental contradiction in the Hilbert program's approach to proof and consistency, revealing inherent limitations similar to Gödel's incompleteness theorems.

Contribution

It shows that the Hilbert program's method for proving consistency leads to a self-referential contradiction, challenging its foundational assumptions.

Findings

Hilbert's approach implies 1-consistency in number-theoretic systems.

Self-referential statements about all $orall_{2}^{0}$ statements lead to contradictions.

The contradiction arises naturally from the Hilbert program's core assumptions.

Abstract

The Hilbert program was actually a specific approach for proving consistency. Quantifiers were supposed to be replaced by $\epsilon$-terms. $\epsilon{x}A(x)$ was supposed to denote a witness to $\exists{x}A(x)$, arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system $S$, each $\epsilon$-term can be replaced by a numeral, making each line provable and true. This implies that $S$ must not only be consistent, but also 1-consistent ($\Sigma_{1}^{0}$-correct). Here we show that if the result is supposed to be provable within $S$, a statement about all $\Pi_{2}^{0}$ statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles G\"odel's but arises naturally out of the Hilbert program itself.