nLab
zeroth-order set theory

Contents

Context

Foundations

Category theory

Topos theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

Zeroth-order set theory is that part of structural set theory that only deals with sets. Compare with first-order set theory, which have families, and higher-order set theory, which have families of families. Also compare with propositional logic, which is logic that deals with only propositions, without any predicates (families of propositions).

Like how intuitionistic propositional logic is the internal logic of an elementary (0,1)-topos or a Heyting algebra and classical propositional logic is the internal logic of a Boolean algebra, constructive zeroth-order set theory is the internal set theory of an elementary (1,1)-topos and classical set theory is the internal set theory of a Boolean topos.

Most structural set theories, such as ETCS or Mike Shulman‘s SEAR, are zeroth-order set theories, as the concept of family is not formalised in the theory.

Created on March 2, 2021 at 23:48:05. See the history of this page for a list of all contributions to it.