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

TL;DR

This paper proves that the Well-Ordering theorem is independent of Ackermann's choice axioms within second-order Henkin logic, providing detailed technical proof in the context of Henkin-Asser structures.

Contribution

It offers a detailed proof of the independence of WO from Ackermann axioms in second-order Henkin predicate logic, extending prior discussions.

Findings

WO is independent of Ackermann axioms in HPL

Provides detailed technical proof of independence

Clarifies the role of Henkin-Asser structures

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). Our discussion is restricted to so-called Henkin-Asser structures of second order. Here, we give the technical details of our proof of the independence of WO from the so-called Ackermann axioms in HPL presented at the Colloquium Logicum in 2022. Most of the definitions used here can be found in Sections 1, 2, and 3 of Part I.