Boolean-valued models and independence proofs in set theory

SET THEORY An Introduction to Independence Proofs

Generalized Algebra-Valued Models of Set Theory

We generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory. §

Orthomodular-Valued Models for quantum Set Theory

Orthomodular logic represented by a complete orthomodular lattice has been studied as a pertinent generalization of the two-valued logic, Boolean-valued logic, and quantum logic. In this paper, we introduce orthomodular logic valued models for set theory generalizing quantum logic valued models introduced by Takeuti as well as Boolean-valued models introduced by Scott and Solovay, and prove a g...

Boolean-valued Term Models

We construct Boolean-valued term models and describe their universal properties. The underlying set of these models consists of the terms of the underlying language, while formulae are interpreted in the Lindenbaum-Tarski algebra of open formulae. Subspaces of the Stone space of the Lindenbaum-Tarski algebra of open formulae are shown to be homeomorphic to spaces of enumerated models models equ...

Boolean-Valued Models and Forcing

This introduction to forcing is based on Chapters 5–6 in T. J. Jech’s book The Axiom of Choice and is written primarily for the Fraenkel-Mostowski Models reading group. Forcing is introduced via Boolean-valued models and generic extensions, and these techniques are used to prove the independence of the axiom of choice of ZF. This is then related back to the theory of FM models. Please inform me...