# Generalized algebra-valued models of set theory

TL;DR: A model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory is shown.

Abstract: We generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory.

## Summary (1 min read)

### Reasonable implication algebras.

- These facts are being used in the calculations later in the paper.
- It is easy to check that all Boolean algebras and Heyting algebras are reasonable and deductive implication algebras.
- The following two examples will be crucial during the rest of the paper:.

### 3.1. Definitions and basic properties.

- In the Boolean-valued case, the names behave nicely with respect to their interpretations as names for sets.
- If two names denote the same object, then the properties of the object do not depend on the name you are using.

### Negation and paraconsistency.

- This definition, together with minimal requirements, makes it impossible to have paraconsistency.
- This gives us the following result immediately: THEOREM 6.1.
- In particular, the class of names x such that N (x) does not form a ∼-equivalence class.
- The authors paraconsistent set theory behaves very differently from the considerations of paraconsistent set theory in the mentioned papers, as the authors can show that the axiom scheme of Comprehension is not valid in their model: THEOREM 6.3.

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### Cites background from "Generalized algebra-valued models o..."

...In recent years, Weber [36, 37], Brady [4], Löwe and Tarafder [21] and others have done exciting work in exploring the possibility of developing a nontrivial set theory built over a paraconsistent logic....

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...In recent years, Weber [36, 37], Brady [4], Löwe and Tarafder [21] and others have done exciting work in exploring the possibility of developing a nontrivial set theory built over a paraconsistent logic....

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..., Libert [20], Löwe and Tarafder [21])....

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...However, there is still no well established model theory for paraconsistent set theory, despite some promising recent developments (see e.g., Libert [20], Löwe and Tarafder [21])....

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...[21] Löwe, B. & Tarafder, S. (2015)....

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6 citations

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##### References

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### "Generalized algebra-valued models o..." refers methods in this paper

...This idea was further generalized by Takeuti & Titani (1992), Titani (1999), Titani & Kozawa (2003), Ozawa (2007), and Ozawa (2009), replacing the Heyting algebra H by appropriate lattices that allow models of quantum set theory (where the algebra is an algebra of truth-values in quantum logic) or…...

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...In the Boolean and Heyting cases, as well as in the algebras considered by Takeuti & Titani (1992), Titani (1999), Titani & Kozawa (2003), and Ozawa (2007, 2009), negation is defined in terms of implication via a∗ := a ⇒ 0....

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84 citations

### "Generalized algebra-valued models o..." refers background in this paper

...The proofs of the Boolean case transfer to the Heyting-valued case to yield that VH is a model of IZF, intuitionistic ZF, where the logic of the Heyting algebra H determines the logic of the Heyting-valued model of set theory (cf. Grayson, 1979; Bell, 2005, chap....

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77 citations

### "Generalized algebra-valued models o..." refers background or methods in this paper

...…the axioms and axiom schemes that we use in our proofs (in the schemes, ϕ is a formula with n + 2 free variables); the concrete formulations follows Bell (2005) very closely: ∀x∀y[∀z(z ∈ x ↔ z ∈ y) → x = y] (Extensionality) ∀x∀y∃z∀w(w ∈ z ↔ (w = x ∨ w = y)) (Pairing) ∃x[∃y(∀z(z ∈ y → ⊥) ∧ y ∈ x)…...

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...Using the notion of validity derived from · , all of the axioms of ZFC are valid in VB. Boolean-valued models were introduced in the 1960s by Scott, Solovay, and Vopěnka; an excellent exposition of the theory can be found in Bell (2005)....

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...Our construction follows very closely the Boolean-valued construction as it can be found in Bell (2005)....

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...In the Boolean case, the inequality proved in Proposition 3.2 is an equality (Bell, 2005, p. 23): ∃x ∈ u ϕ(x) = ∨ x∈dom(u) ( u(x) ∧ ϕ(x) ) and ∀x ∈ u ϕ(x) = ∧ x∈dom(u) ( u(x) ⇒ ϕ(x) ) ....

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...As in the Boolean case (Bell, 2005, Induction Principle 1.7), the (meta-)induction principle for VA can be proved by a simple induction on the rank function: for every property of names, if for all x ∈ VA, we have ∀y ∈ dom(x)( (y)) implies (x), then all names x ∈ VA have the property ....

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