Sheaf models for set theory

The associated sheaf functor theorem in algebraic set theory

We prove a version of the associated sheaf functor theorem in Algebraic Set Theory. The proof is established working within a Heyting pretopos equipped with a system of small maps satisfying the axioms originally introduced by Joyal and Moerdijk. This result improves on the existing developments by avoiding the assumption of additional axioms for small maps and the use of collection sites.

Sheaf models for choice sequences

Models of Set Theory

1. First order logic and the axioms of set theory 2 1.1. Syntax 2 1.2. Semantics 2 1.3. Completeness, compactness and consistency 3 1.4. Foundations of mathematics and the incompleteness theorems 3 1.5. The axioms 4 2. Review of basic set theory 5 2.1. Classes 5 2.2. Well-founded relations and recursion 5 2.3. Ordinals, cardinals and arithmetic 6 3. The consistency of the Axiom of Foundation 8 ...

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...

MODELS OF SET THEORY 1 Models