Ordinals in an Algebra-Valued Model of a Paraconsistent Set Theory
08 Jan 2015-pp 195-206
TL;DR: 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.
Abstract: 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.
Citations
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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.
24 citations
Cites background from "Ordinals in an Algebra-Valued Model..."
...A first discussion of the behaviour of von Neumann ordinals in V(PS3, ∗) can be found in Tarafder (2015)....
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TL;DR: Modifications of Semmes's game characterization of the Borel functions are defined, obtaining game characterizations of the Baire class $\alpha$ functions for each fixed $\alpha < \omega_1$.
Abstract: 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.
18 citations
01 Jan 2015
TL;DR: Soundness and completeness theorems are established in a three-valued paraconsistent logic obtained from some algebra-valued model of set theory.
Abstract: 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.
8 citations
Posted Content•
TL;DR: It is argued 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 .
Abstract: 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).
7 citations
Cites background from "Ordinals in an Algebra-Valued Model..."
...As discussed in Section 9, the three-valued logic (PS3, ∗) used in [22] already appears in [10] under the name MPT, and it is equivalent to LPT0 and also to LFI1◦....
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TL;DR: In this paper, the authors show 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.
Abstract: 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.
6 citations
References
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TL;DR: An enormous variety of paraconsistent logics in the literature is shown to constitute C- System, and a novel notion of consistency is introduced.
Abstract: A thorough investigation of the foundations of paraconsistent logics. Relations between logical principles are formally studied, a novel notion of consistency is introduced, the logics of formal inconsistency, and the subclasses of C-systems and dC-systems are defined and studied. An enormous variety of paraconsistent logics in the literature is shown to constitute C-systems.
185 citations
01 Jan 1979
86 citations
"Ordinals in an Algebra-Valued Model..." refers methods in this paper
...If the complete Boolean algebra is replaced by a complete Heyting algebra, H then essentially the same proof shows that V becomes a model of Intuitionistic Zermelo Fraenkel set theory, IZF [4]....
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...This fact is corroborated in Intuitionistic set theory as viewed in Heyting algebra-valued models [4]....
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Book•
17 Dec 2011
TL;DR: In this paper, the authors present a list of problems with Boolean and Heyting algebra-valued models: first steps, forcing and some independent proofs, group actions on V(B) and the Independence of the Axiom of Choice, generic Ultrafilters and Transitive Models of ZFC.
Abstract: Foreword Preface List of Problems 0 Boolean and Heyting Algebras: The Essentials 1 Boolean-Valued Models: First Steps 2 Forcing and Some Independece Proofs 3 Group Actions on V(B) and the Independence of the Axiom of Choice 4 Generic Ultrafilters and Transitive Models of ZFC 5 Cardinal Collapsing, Boolean Isomorphism and Applications to the Theory of Boolean Algebras 6 Iterated Boolean Extensions, Martin's Axiom and Souslin's Hypothesis 7 Boolean-Valued Analysis 8 Intuitionistic Set Theory and Heyting-Algebra-Valued Models Appendix Boolean- and Heyting-Algebra-Valued Models as Categories Historical Notes Bibliography Index of Symbols Index of Terms
85 citations
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TL;DR: This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic, and indicates how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.
Abstract: This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.
69 citations
"Ordinals in an Algebra-Valued Model..." refers background in this paper
...On the other hand it is a theorem of the paraconsistent set theory considered in [12]....
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...On the other hand let us consider the paraconsistenet set theory described by [12] where the classical definition of ordinal is used....
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