Functional Representation of a Boolean Valued Universe

Gutman, A. E.; Losenkov, G. A.

doi:10.1007/978-94-011-4305-9_2

Cited by 2 publications

(3 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(a) If X satisfies the mixing principle, then X satisfies the maximum principle (see [9] (1.10), [5] (6.1.7)).…”

Section: The Maximum Principlementioning

confidence: 99%

See 1 more Smart Citation

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

Gutman

2021

Mathematics

View full text Add to dashboard Cite

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.

show abstract

“…(a) If X satisfies the mixing principle, then X satisfies the maximum principle (see [9] (1.10), [5] (6.1.7)).…”

Section: The Maximum Principlementioning

confidence: 99%

“…(b) If X is intensional and satisfies the maximum principle for the formula (∃ x)(x ∈ y), then X satisfies the mixing principle (see [9] (1.12), [5] (6.1.9)). (c) If X is extensional and satisfies the ascent principle, then X satisfies the mixing and maximum principles (see [9] (1.11), [5] (6.1.8)).…”

Section: The Maximum Principlementioning

confidence: 99%

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

Gutman

2021

Mathematics

View full text Add to dashboard Cite

show abstract

“…This universal construction belongs to Gutman and Losenkov (cp. [7]). The functional realization visualizes descending and ascending, the Escher rules, and the Gordon Theorem (cp.…”

mentioning

confidence: 99%

Leibnizian, Robinsonian, and Boolean valued monads

Kutateladze

2011

J. Appl. Ind. Math.

View full text Add to dashboard Cite

Abstract-This is an overview of the present-day versions of monadology with some applications to vector lattices and linear inequalities. Two approaches to combining nonstandard set-theoretic models are sketched and illustrated by order convergence, principal projection, and polyhedrality. DOI: 10.1134/S1990478911030082Keywords: Dedekind complete vector lattice, filter, fragments, principal projection, order convergence, up-down, descent, ascent, polyhedral Lagrange principle, Boolean valued model The notion of monad is central to external set theory. Justifying the simultaneous use of infinitesimals and the technique of descending and ascending in vector lattice theory requires adaptation of monadology for the implementation of filters in Boolean valued universes. This is still a rather uncharted area of research. The two approaches are available now. One is to apply monadology to the descents of objects. The other consists in applying the standard monadology inside the Boolean valued universe V (B) over a complete Boolean algebra B, while ascending and descending by the Escher rules (cp.[1] and [2]).These approaches are sketched and illustrated by tests for order convergence and rules for fragmenting and projecting positive operators in vector lattices. Also, Lagrange's principle is shortly addressed in polyhedral environment with inexact data.

show abstract

Functional Representation of a Boolean Valued Universe

Cited by 2 publications

References 2 publications

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

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

Leibnizian, Robinsonian, and Boolean valued monads

Contact Info

Product

Resources

About