Relationships between Principles of Choice in Second-Order Henkin Structures

TL;DR

This paper explores the strength of second-order versions of the Axiom of Choice within Henkin semantics, establishing relationships between various axioms in second-order predicate logic.

Contribution

It provides new proofs linking Ackermann, Zermelo-Asser, and Asser axioms in the context of second-order Henkin structures, expanding understanding of choice principles.

Findings

Established relationships between Ackermann and Zermelo-Asser axioms.

Proved connections between Asser and Zermelo-Asser axioms.

Detailed proofs of these relationships presented.

Abstract

We deal with the strength of classical second-order versions of the Axiom of Choice (AC) in second-order predicate logic (PLII) with Henkin interpretation (HPL). We use the known relationships between the so-called Zermelo-Asser axioms and the so-called Russell-Asser axioms and prove relationships between the so-called Ackermann axioms and the Zermelo-Asser axioms and between the so-called Asser axioms and the Zermelo-Asser axioms. In particular, we give the technical details of the proofs of our results presented at the DMV Annual Meeting 2022.