THE COLLAPSE OF THE HILBERT PROGRAM: A VARIATION ON THE GÖDELIAN THEME

Abstract

The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ϵ-terms. ϵxA(x) was supposed to denote a witness to, or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ϵ-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 (-correct). Here we show that if the result is supposed to be provable within S, a statement about all statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel's but arises naturally out of the Hilbert program itself.