Boolean-Valued Universe as an Algebraic System. II: Intensional Hierarchies

Gutman, A. E.

doi:10.1134/s0037446620030052

Cited by 1 publication

(2 citation statements)

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“…References are also provided for the author's publications, with the following exception. The present article continues, develops, and supersedes the study initiated in [15][16][17]. The content of those three papers has been redesigned, corrected, extended, and distributed across the new subsections.…”

Section: Discussionmentioning

confidence: 84%

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Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique

Gutman

2021

Mathematics

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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 Booleanvalued 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.

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Section: Discussionmentioning

confidence: 84%

“…(Moreover, [15] is available only in Russian.) Therefore, we confine ourselves to including [15][16][17] in the list of references.…”

Section: Discussionmentioning

confidence: 99%