Generalized algebra-valued models of set theoryLöwe, B.; Tarafder, S. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. 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. §1. Introduction. If B is any Boolean algebra and V a model of set theory, we can construct by transfinite recursion the Boolean-valued model of set theory V B consisting of names for sets, an extended language L B , and an interpretation function · : L B → B assigning truth values in B to formulas of the extended language. Using the notion of validity derived from · , all of the axioms of ZFC are valid in V B . 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).Replacing the Boolean algebra in the above construction by a Heyting algebra H, one obtains a Heyting-valued model of set theory V H . The proofs of the Boolean case transfer to the Heyting-valued case to yield that V H 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. 8). 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 fuzzy set theory.In this paper, we shall generalize this model construction further to work on algebras that we shall call reasonable implication algebras ( §2). These algebras do not have a negation symbol, and hence we shall be focusing on the negation-free fragment of first-order logic: the closure under the propositional connectives ∧, ∨, ⊥, and →. Classically, of course, every formula is equivalent to one in the negation-free fragment (since ¬ϕ is equivalent to ϕ → ⊥). In §3, we define the model construction and prove that assuming a number of additional assumptions (among them a property we call the bound...

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An algebra-valued model of set theory is called loyal to its algebra if the model and its algebra have the same propositional logic; it is called faithful if all elements of the algebra are truth values of a sentence of the language of set theory in the model. We observe that non-trivial automorphisms of the algebra result in models that are not faithful and apply this to construct three classes of illoyal models: tail stretches, transposition twists, and maximal twists.