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Interpretations of ZF

Santiago Jockwich, Sourav Tarafder, Giorgio Venturi

Abstract

In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for identity and set-membership. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of ZF to new algebra-valued models.

Interpretations of ZF

Abstract

In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for identity and set-membership. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of ZF to new algebra-valued models.
Paper Structure (34 sections, 32 theorems, 103 equations, 1 figure, 2 tables)

This paper contains 34 sections, 32 theorems, 103 equations, 1 figure, 2 tables.

Table of Contents

  1. Algebra-valued models
  2. Algebraic structures
  3. Assignment functions and validity of a sentence
  4. Assignment functions with two different interpretations for $\in$ and $=$
  5. The assignment function $\llbracket \cdot \rrbracket_\mathrm{BA}$ and $\mathsf{ZF}$
  6. Boolean and Heyting-valued models
  7. Generalized algebra-valued models
  8. $\mathcal{T}$-valued models
  9. Designated $\mathsf{Cobounded}$-algebra
  10. Examples of designated $\mathsf{Cobounded}$-algebras
  11. Main goal of the paper
  12. Cobounded-algebra-valued models under $\llbracket \cdot \rrbracket_\mathrm{BA}$ and $\llbracket \cdot \rrbracket_\mathrm{PA}$
  13. The algebra $\mathbb{PS}_3$
  14. $\mathsf{Cobounded}$-algebra-valued models under $\llbracket \cdot \rrbracket_\mathrm{BA}$
  15. Set-theoretic properties of $\mathsf{Cobounded}$-algebra-valued models under $\llbracket \cdot \rrbracket_\mathrm{PA}$
  16. ...and 19 more sections

Key Result

Theorem 2.1

Let $\mathbb{A}$ be any complete Boolean algebra and $\{\mathbf{1}\}$ be the designated set, where $\mathbf{1}$ is the top element of $\mathbb{A}$, then $\mathbf{V}^{(\mathbb{A}, ~ \llbracket \cdot \rrbracket_\mathrm{BA})} \models \mathsf{ZF}$.In this paper, we will not deal with the Axiom of Choice

Figures (1)

  • Figure 1: The axioms of $\mathsf{ZF}$.

Theorems & Definitions (78)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 2.1: cf. Bell, Theorem 1.33
  • Theorem 2.2: grayson, p. 410
  • Definition 2.3: Loewe, p. 194
  • ...and 68 more