The Explicit Definition of Quantifiers via Hilbert's epsilon is Confluent and Terminating (1611.06389v1)

Abstract: We investigate the elimination of quantifiers in first-order formulas via Hilbert's epsilon-operator (or -binder), following Bernays' explicit definitions of the existential and the universal quantifier symbol by means of epsilon-terms. This elimination has its first explicit occurrence in the proof of the first epsilon-theorem in Hilbert-Bernays in 1939. We think that there is a lacuna in this proof w.r.t. this elimination, related to the erroneous assumption that explicit definitions always terminate. Surprisingly, to the best of our knowledge, nobody ever proved confluence or termination for this elimination procedure. Even myths on non-confluence and the openness of the termination problem are circulating. We show confluence and termination of this elimination procedure by means of a direct, straightforward, and easily verifiable proof, based on a new theorem on how to obtain termination from weak normalization.