Herbrand complexity and the epsilon calculus with equality

Abstract

The \(\varepsilon \)-elimination method of Hilbert’s \(\varepsilon \)-calculus yields the up-to-date most direct algorithm for computing the Herbrand disjunction of an extensional formula. A central advantage is that the upper bound on the Herbrand complexity obtained is independent of the propositional structure of the proof. Prior (modern) work on Hilbert’s \(\varepsilon \)-calculus focused mainly on the pure calculus, without equality. We clarify that this independence also holds for first-order logic with equality. Further, we provide upper bounds analyses of the extended first \(\varepsilon \)-theorem, even if the formalisation incorporates so-called \(\varepsilon \)-equality axioms.