An abstract form of the first epsilon theorem

Abstract We present a new method of computing Herbrand disjunctions. The up-to-date most direct approach to calculate Herbrand disjunctions is based on Hilbert’s epsilon formalism (which is in fact also the oldest framework for proof theory). The algorithm to calculate Herbrand disjunctions is an integral part of the proof of the extended first epsilon theorem. This paper introduces a more abstract form of epsilon proofs, the function variable proofs. This leads to a computational improved version of the extended first epsilon theorem, which allows a nonelementary speed up of the computation of Herbrand disjunctions. As an application, sequent calculus proofs are translated into function variable proofs and a variant of the axiom of global choice is shown to be removable from proofs in Neumann–Bernays–Gödel set theory.