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Mathematics
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Boolean Valued Analysis with Applications
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Boolean Valued Analysis with Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (1 March 2021) | Viewed by 8371

Special Issue Editors

Prof. Dr. Anatoly Georgievich Kusraev

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Guest Editor
Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Russia
Interests: Boolean valued analysis; vector lattices; operator theory; convex analysis
Prof. Dr. Semën Kutateladze

E-Mail Website
Guest Editor
Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences, Russia
Interests: nonstandard methods of analysis; vector lattices; operator theory; convex geometry; optimization

Special Issue Information

Dear Colleagues,

Boolean valued analysis signifies the technique of studying the properties of an arbitrary mathematical object by comparison between its representations in two different set-theoretic models whose construction utilizes principally distinct Boolean algebras. These models are usually the classical Cantorian paradise in the shape of the von Neumann universe and a specially-trimmed Boolean valued universe. Comparison analysis is carried out via some interplay between the universes.

The purpose of this Special Issue is to gather a collection of articles reflecting the latest developments in various sections of operator theory, measure and integration, operator algebras, convex analysis, mathematical finance, etc. The collection is intended for the classical analyst seeking new powerful tools and for the model theorist in search of challenging applications of nonstandard models of set theory.

Prof. Dr. Anatoly Georgievich Kusraev
Prof. Dr. Semën Kutateladze
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

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Keywords

  • Boolean valued model
  • Boolean truth value
  • Ascents and descents
  • Transfer principle
  • Boolean valued reals
  • Cardinal collapsing
  • Vector lattice
  • Positive operator
  • Measure algebra
  • Projection algebra

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Published Papers (4 papers)

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78 pages, 660 KiB  
Article
Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique
by Alexander Gutman
Mathematics 2021, 9(9), 1056; https://doi.org/10.3390/math9091056 - 8 May 2021
Cited by 2 | Viewed by 1616
Abstract
This article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols, internal [...] Read more.
This article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols, internal classes, outer terms and external Boolean-valued classes. Next, we enrich the collection of Boolean-valued research tools with the technique of partial elements and the corresponding joins, mixings and ascents. Passing on to the set-theoretic signature, we prove that bounded formulas are absolute for transitive Boolean-valued subsystems. We also introduce and study intensional, predicative, cyclic and regular Boolean-valued systems, examine the maximum principle, and analyze its relationship with the ascent and mixing principles. The main applications relate to the universe over an arbitrary extensional Boolean-valued system. A close interrelation is established between such a universe and the intensional hierarchy. We prove the existence and uniqueness of the Boolean-valued universe up to a unique isomorphism and show that the conditions in the corresponding axiomatic characterization are logically independent. We also describe the structure of the universe by means of several cumulative hierarchies. Another application, based on the quantifier hierarchy of formulas, improves the transfer principle for the canonical embedding in the Boolean-valued universe. Full article
(This article belongs to the Special Issue Boolean Valued Analysis with Applications)
10 pages, 268 KiB  
Article
From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti
by Masanao Ozawa
Mathematics 2021, 9(4), 397; https://doi.org/10.3390/math9040397 - 17 Feb 2021
Cited by 1 | Viewed by 2353
Abstract
Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the [...] Read more.
Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of the Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory that is based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti’s mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years. Full article
(This article belongs to the Special Issue Boolean Valued Analysis with Applications)
18 pages, 402 KiB  
Article
Geometric Characterization of Injective Banach Lattices
by Anatoly Kusraev and Semën Kutateladze
Mathematics 2021, 9(3), 250; https://doi.org/10.3390/math9030250 - 27 Jan 2021
Cited by 1 | Viewed by 1755
Abstract
This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach [...] Read more.
This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an AL-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding AL-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices. Full article
(This article belongs to the Special Issue Boolean Valued Analysis with Applications)
23 pages, 396 KiB  
Article
Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations
by Antonio Avilés López and José Miguel Zapata García
Mathematics 2020, 8(10), 1848; https://doi.org/10.3390/math8101848 - 20 Oct 2020
Viewed by 1972
Abstract
We establish a connection between random set theory and Boolean valued analysis by showing that random Borel sets, random Borel functions, and Markov kernels are respectively represented by Borel sets, Borel functions, and Borel probability measures in a Boolean valued model. This enables [...] Read more.
We establish a connection between random set theory and Boolean valued analysis by showing that random Borel sets, random Borel functions, and Markov kernels are respectively represented by Borel sets, Borel functions, and Borel probability measures in a Boolean valued model. This enables a Boolean valued transfer principle to obtain random set analogues of available theorems. As an application, we establish a Boolean valued transfer principle for large deviations theory, which allows for the systematic interpretation of results in large deviations theory as versions for Markov kernels. By means of this method, we prove versions of Varadhan and Bryc theorems, and a conditional version of Cramér theorem. Full article
(This article belongs to the Special Issue Boolean Valued Analysis with Applications)
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