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Register a new userExtensional realizability for intuitionistic set theory
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In generic realizability for set theories, realizers treat unbounded quantifiers generically. To this form of realizability, we add another layer of extensionality by requiring that realizers ought to act extensionally on realizers, giving rise to a realizability universe $mathrm{V_{ex}}(A)$ in which the axiom of choice in all finite types ${sf AC}_{{sf FT}}$ is realized, where $A$ stands for an arbitrary partial combinatory algebra. This construction furnishes inner models of many set theories that additionally validate ${sf AC}_{{sf FT}}$, in particular it provides a self-validating semantics for $sf CZF$ (Constructive Zermelo-Fraenkel set theory) and $sf IZF$ (Intuitionistic Zermelo-Fraenkel set theory). One can also add large set axioms and many other principles.
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Goodmans theorem (1976) states that intuitionistic finite-type arithmetic plus the axiom of choice plus the axiom of relativized dependent choice is conservative over Heyting arithmetic. The same result applies to the extensional variant. This is due to Beeson (1979). In this paper we modify Goodman realizability (1978) and provide a new proof of the extensional case.
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--Godel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science.
We first show that in the function realizability topos every metric space is separable, and every object with decidable equality is countable. More generally, working with synthetic topology, every $T_0$-space is separable and every discrete space is countable. It follows that intuitionistic logic does not show the existence of a non-separable metric space, or an uncountable set with decidable equality, even if we assume principles that are validated by function realizability, such as Dependent and Function choice, Markovs principle, and Brouwers continuity and fan principles.
In this paper we prove that Neutrosophic Set (NS) is an extension of Intuitionistic Fuzzy Set (IFS) no matter if the sum of single-valued neutrosophic components is < 1, or > 1, or = 1. For the case when the sum of components is 1 (as in IFS), after applying the neutrosophic aggregation operators one gets a different result from that of applying the intuitionistic fuzzy operators, since the intuitionistic fuzzy operators ignore the indeterminacy, while the neutrosophic aggregation operators take into consideration the indeterminacy at the same level as truth-membership and falsehood-nonmembership are taken. NS is also more flexible and effective because it handles, besides independent components, also partially independent and partially dependent components, while IFS cannot deal with these. Since there are many types of indeterminacies in our world, we can construct different approaches to various neutrosophic concepts. Also, Regret Theory, Grey System Theory, and Three-Ways Decision are particular cases of Neutrosophication and of Neutrosophic Probability. We extended for the first time the Three-Ways Decision to n-Ways Decision, and the Spherical Fuzzy Set to n-HyperSpherical Fuzzy Set and to n-HyperSpherical Neutrosophic Set.
Descriptive set theory was originally developed on Polish spaces. It was later extended to $omega$-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending descriptive set theory and its effective counterpart to general represented spaces, including non-countably-based spaces has been started in [Pauly, de Brecht 2015]. We study the spaces $mathcal{O}(mathbb{N}^mathbb{N})$, $mathcal{C}(mathbb{N}^mathbb{N},2)$ and the Kleene-Kreisel spaces $mathbb{N}langlealpharangle$. We show that there is a $Sigma^0_2$-subset of $mathcal{O}(mathbb{N}^mathbb{N})$ which is not Borel. We show that the open subsets of $mathbb{N}^{mathbb{N}^mathbb{N}}$ cannot be continuously indexed by elements of $mathbb{N}^mathbb{N}$ or even $mathbb{N}^{mathbb{N}^mathbb{N}}$, and more generally that the open subsets of $mathbb{N}langlealpharangle$ cannot be continuously indexed by elements of $mathbb{N}langlealpharangle$. We also derive effecti
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