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

Christine Gaßner

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.

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

Paper Structure

This paper contains 16 sections, 13 theorems, 40 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Finite, infinite, basic, and standard structures
  3. The models used
  4. Finite stabilizers
  5. Choice functions and finite families
  6. Why is the basic Fraenkel model a model of HAC?
  7. A list of ideas for proving HAC
  8. $P$-adequate decompositions
  9. The description of the consequences of swapping two individuals
  10. The 1-1 Ackermann axioms hold in the basic model
  11. HAC, TR, and a finite union of Fraenkel models
  12. The characterization of a finite union of Henkin structures
  13. The current list of ideas for proving HAC
  14. The 1-1 Ackermann axioms hold in a union of Fraenkel models
  15. Consequences: The independence of TR from AC
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $H( x ,D)$ be in ${\cal L}^{(2)}_{ x ,D}$. Let $\Sigma$ be any structure of the form $\Sigma(I,{\mathfrak{G}},{\cal I}_{\sf 0}^I)$ for any subgroup $\mathfrak{G}\subseteq {\mathfrak{G}}_{\sf 1}^I$ and let $f$ be in ${\rm assgn}(\Sigma)$. Then, there is a finite set $P$ in ${\cal I}_{\sf 0}^I$ su holds for all $\xi\in I$, for all predicates $\delta\in J_1(I,{\mathfrak{G}},{\cal I}_{\sf 0}^I)$,

Theorems & Definitions (15)

  • Lemma 2.1: Finite supports and stabilizers for formulas
  • Lemma 2.2: The assumption $\forall x \exists D H( x ,D)$ and choice functions
  • Lemma 2.3: $choice_h^{1,1}(H)$ restricted to finite families $({\cal A}^H_{\xi})_{ \xi \in \alpha}$
  • Proposition 2.4: $choice_h^{(2)}\!$ and $WO^n$ hold in finite Henkin structures
  • Remark 2.5: The ${\rm HPL}$-independence of $WO^n$
  • Remark 2.6: Weaker assumptions
  • Lemma 3.2: $P$-adequate partition of an individual domain
  • Lemma 3.3: Formulas describing the $P$-adequate partition of $I$
  • Lemma 3.4: A choice set for the $P$-adequate partition of $I$
  • Lemma 3.5: A formula for permuting individuals
  • ...and 5 more