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

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.