#
nLab

zeroth-order set theory

### Context

#### Foundations

**foundations**

## The basis of it all

## Foundational axioms

## Removing axioms

#### Category theory

**category theory**

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

#### Topos theory

**topos theory**

## Background

## Toposes

## Internal Logic

## Topos morphisms

## 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.