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

TL;DR

This paper investigates the independence of the well-ordering theorem from Ackermann's choice principle within second-order Henkin logic, providing technical details for a proof of their independence.

Contribution

It offers a detailed proof of the independence of WO from Ackermann's axioms in second-order Henkin predicate logic, expanding understanding of foundational logical principles.

Findings

Proved the independence of WO from Ackermann axioms in HPL.

Provided technical details for the independence proof.

Clarified the role of Henkin-Asser structures in second-order logic.

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.