Semantics and Proof Theory of the Epsilon Calculus

TL;DR

This paper explores the semantics and proof theory of the epsilon calculus, addressing its proof systems and semantic interpretations to enhance its formal understanding and application.

Contribution

It provides an analysis of the epsilon calculus's proof systems, discusses semantic interpretations, and outlines challenges in developing well-behaved proof-theoretic systems.

Findings

Presentation of the first epsilon theorem.

Discussion of the epsilon calculus proof system.

Outline of challenges in proof-theoretic development.

Abstract

The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. The application of this undervalued formalism has been hampered by the absence of well-behaved proof systems on the one hand, and accessible presentations of its theory on the other. One significant early result for the original axiomatic proof system for the epsilon-calculus is the first epsilon theorem, for which a proof is sketched. The system itself is discussed, also relative to possible semantic interpretations. The problems facing the development of proof-theoretically well-behaved systems are outlined.