Generalized algebra-valued models of 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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"Generalized algebra-valued models o..." refers background in this paper
...Paraconsistent set theories have been studied by many authors (Brady, 1971; Brady & Routley, 1989; Restall, 1992; Libert, 2005; Weber, 2010a,b, 2013); all of these accounts start from the observation that ZF was created to avoid the contradiction that can be obtained from the axiom scheme of…...
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"Generalized algebra-valued models o..." refers background in this paper
...…ϕ(x) → ϕ(y)) 3 This implication-negation algebra was introduced by Marcos (2000) as one of the 8,192 maximal paraconsistent three-valued logics mentioned in the title of the paper; it was further studied in Carnielli & Marcos (2002, § 3.11), Marcos (2005), and Coniglio & da Cruz Silvestrini (2014)....
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...As discussed in §4, the natural approach here would be consider the ∼-equivalence classes of names as objects where u ∼ v if and only if u = v ∈ D.4 Due to the proof of Theorem 6.1, we cannot expect that (the scheme of) Leibniz’s Law ∀x∀y(x = y ∧ ϕ(x) → ϕ(y)) 3 This implication-negation algebra was introduced by Marcos (2000) as one of the 8,192 maximal paraconsistent three-valued logics mentioned in the title of the paper; it was further studied in Carnielli & Marcos (2002, § 3.11), Marcos (2005), and Coniglio & da Cruz Silvestrini (2014)....
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28 citations
"Generalized algebra-valued models o..." refers background in this paper
...Paraconsistent set theories have been studied by many authors (Brady, 1971; Brady & Routley, 1989; Restall, 1992; Libert, 2005; Weber, 2010a,b, 2013); all of these accounts start from the observation that ZF was created to avoid the contradiction that can be obtained from the axiom scheme of Comprehension ∃x∀y(y ∈ x ↔ φ(y))...
[...]
...Paraconsistent set theories have been studied by many authors (Brady, 1971; Brady & Routley, 1989; Restall, 1992; Libert, 2005; Weber, 2010a,b, 2013); all of these accounts start from the observation that ZF was created to avoid the contradiction that can be obtained from the axiom scheme of…...
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