Naive set theory

Definition

Naive set theory, in its formal reading, is the set theory in first-order logic with the relation symbol $\in$ and these two axioms.

Extensionality: two sets with the same elements are equal, as in the axiom of extensionality.
Unrestricted comprehension: for each formula $\varphi(x)$ , possibly with parameters, in which $y$ is not free, $\exists y\, \forall x\, (x \in y \leftrightarrow \varphi(x))$ .

In a wider sense, naive set theory is the informal use of sets in ordinary language, without a fixed list of axioms.

Remarks

Example: the comprehension instance for $\varphi(x) = (x \ne x)$ gives a set with no elements, the empty set, and extensionality makes it unique. Non-example: the formal theory is inconsistent, so it has no model. Indeed, take $\varphi(x) = (x \notin x)$ ; comprehension gives $y$ with $\forall x\, (x \in y \leftrightarrow x \notin x)$ , and $x = y$ gives $y \in y \leftrightarrow y \notin y$ . This is Russell's paradox. Zermelo–Fraenkel set theory replaces unrestricted comprehension by the separation scheme $\exists y\, \forall x\, (x \in y \leftrightarrow x \in a \wedge \varphi(x))$ , which only forms subsets of a given set $a$ .

Tag	`5MDSVRCDZW`
URI	`m2://5MDSVRCDZW`
Kind	definition
Writing	W0 stub
MSC	03E30, 03A05
Defines	Naive set theory
Uses	Set theory, Set, Axiom of extensionality, Russell's paradox, Zermelo–Fraenkel set theory
Changed	2 November 2026

Tag

5MDSVRCDZW

URI

m2://5MDSVRCDZW

Kind

definition

Writing

W0 stub

MSC

03E30, 03A05

Defines

Naive set theory

Uses

Set theory, Set, Axiom of extensionality, Russell's paradox, Zermelo–Fraenkel set theory

Changed

2 November 2026

Naive set theory

Definition

Remarks

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