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Register a new userDecidability of the decision problem for Boolean set theory with the unordered Cartesian product operator
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We solve the decidability problem for Boolean Set Theory with unordered cartesian product.
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We relate the decidability problem for BS with unordered cartesian product with Hilberts Tenth problem and prove that BS with unordered cartesian product is NP-complete.
We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prufer (in particular Bezout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For B{e}zout domains these conditions are also necessary.
We prove that the theory of all modules over the ring of algebraic integers is decidable.
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.
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.
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