TY - JOUR

T1 - Boolean-valued set-theoretic systems: General formalism and basic technique

AU - Gutman, Alexander

N1 - Gutman A. Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique. Mathematics. 2021; 9(9):1056. The work was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project 0314?2019?0005).

PY - 2021/5/8

Y1 - 2021/5/8

N2 - 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.

AB - 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.

KW - Algebraic system

KW - Boolean-valued universe

KW - Cumulative hierarchy

KW - Set theory

UR - http://www.scopus.com/inward/record.url?scp=85106581080&partnerID=8YFLogxK

U2 - 10.3390/math9091056

DO - 10.3390/math9091056

M3 - Article

AN - SCOPUS:85106581080

VL - 9

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 9

M1 - 1056

ER -