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.

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Generalized algebra-valued models of set theory

Game characterizations of classes of functions in descriptive set theory have their origins in the seminal work of Wadge, with further developments by several others. In this thesis we study such characterizations from several perspectives. We define modifications of Semmes's game characterization of the Borel functions, obtaining game characterizations of the Baire class $\alpha$ functions for each fixed $\alpha < \omega_1$. We also define a construction of games which transforms a game characterizing a class $\Lambda$ of functions into a game characterizing the class of functions which are piecewise $\Lambda$ on a countable partition by $\Pi^0_\alpha$ sets, for each $0 < \alpha < \omega_1$. We then define a parametrized Wadge game by using computable analysis, and show how the parameters affect the class of functions that is characterized by the game. As an application, we recast our games characterizing the Baire classes into this framework. Furthermore, we generalize our game characterizations of function classes to generalized Baire spaces, show how the notion of computability on Baire space can be transferred to generalized Baire spaces, and show that this is appropriate for computable analysis by defining a representation of Galeotti's generalized real line and analyzing the Weihrauch degree of the intermediate value theorem for that space. Finally, we show how the game characterizations of function classes discussed lead in a natural way to a stratification of each class into a hierarchy, intuitively measuring the complexity of functions in that class. This idea and the results presented open new paths for further research.

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Games for Functions: Baire Classes, Weihrauch Degrees, Transfinite Computations, and Ranks

Paraconsistent logic makes it possible to study inconsistent theories in a coherent way. From its modern start in the mid-20th century, paraconsistency was intended for use in mathematics, providing a rigorous framework for describing abstract objects and structures where some contradictions are allowed, without collapse into incoherence. Over the past decades, this initiative has evolved into an area of non-classical mathematics known as inconsistent or paraconsistent mathematics. This Element provides a selective introductory survey of this research program, distinguishing between `moderate' and `radical' approaches. The emphasis is on philosophical issues and future challenges.

Paraconsistency in Mathematics

This paper presents a three-valued paraconsistent logic obtained from some algebra-valued model of set theory. Soundness and completeness theorems are established. The logic has been compared with other three-valued paraconsistent logics.

A Paraconsistent Logic Obtained from an Algebra-Valued Model of Set Theory

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vop\v{e}nka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, L\"{o}we and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by L\"{o}we and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing L\"{o}we and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF).

Twist-Valued Models for Three-valued Paraconsistent Set Theory

This paper deals with ordinal numbers in an algebra-valued model of a paraconsistent set theory. It is proved that the collection of all ordinals is not a set in this model which is dissimilar to the other existing paraconsistent set theories. For each ordinal α of classical set theory α-like elements are defined in the mentioned algebra-valued model whose collection is not singleton. It is shown that two α-like elements (for same α) may perform conversely to validate a given formula of the corresponding paraconsistent set theory.

Ordinals in an Algebra-Valued Model of a Paraconsistent Set Theory

\n In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of \n \n \n \n$\\mathsf {CH}$\n\n \n ); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse.

Sourav Tarafder

Papers

Generalized algebra-valued models of set theory

A Paraconsistent Logic Obtained from an Algebra-Valued Model of Set Theory

Ordinals in an Algebra-Valued Model of a Paraconsistent Set Theory

Independence proofs in non-classical set theories

Ideal Objects for Set Theory