AC and the Independence of WO in Second-Order Henkin Logic, Part I

Abstract

This article is concerned with the Axiom of Choice (AC) and the well-ordering theorem (WO) in second-order predicate logic with Henkin interpretation (HPL). We consider a principle of choice introduced by Wilhelm Ackermann (1935) and discussed also by David Hilbert and Ackermann (1938), by G\"unter Asser (1981), and by Benjamin Siskind, Paolo Mancosu, and Stewart Shapiro (2020). The discussion is restricted to so-called Henkin-Asser structures of second order. The language used is a many-sorted first-order language with identity. In particular, we give some of the technical details for a proof of the independence of WO from the so-called Ackermann axioms in HPL presented at the Colloquium Logicum in 2022.