AC and the Independence of the Law of Trichotomy in Second-Order Henkin Logic

TL;DR

This paper investigates the independence of the law of trichotomy within second-order Henkin logic, demonstrating its independence from Ackermann axioms and exploring related model-theoretic properties.

Contribution

It establishes the independence of the law of trichotomy from Ackermann axioms in second-order Henkin logic, answering a question by Michael Rathjen.

Findings

The law of trichotomy is independent of Ackermann axioms in HPL.

The basic Fraenkel model satisfies all Ackermann axioms in HAC.

Open problem related to the independence in the logic.

Abstract

This paper focuses on the set HAC of 1-1 Ackermann axioms of choice in second-order predicate logic with Henkin interpretation (HPL). To answer a question posed by Michael Rathjen, we restrict the proof that the basic Fraenkel model of second order is a model of all n-m Ackermann axioms to the case where the Ackermann axioms are in HAC. In the second part, we show the independence of Hartogs' version of the law of trichotomy (TR) from HAC in HPL. A generalization of the latter proof implies the independence of TR from all Ackermann axioms in HPL. We conclude the paper with an open problem.