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Canonical static concept record

Zermelo–Fraenkel set theory (ZF)

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Summary

The standard first-order set theory with extensionality, foundation, pairing, union, power set, infinity, and the separation and replacement schemas, but without the axiom of choice.

Record metadata

Canonical name: Zermelo–Fraenkel set theory (ZF)
Concept ID: zermelo_fraenkel_set_theory
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory, Logic
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Axiomatic theory

Carrier(s)

first-order language with binary relation $\in$

Data

axioms and axiom schemas of ZF

Canonically induces

formal universe in which the set-theoretic constructions are carried out

Notes

Equivalent presentations may trade some axioms for theorems; the defining editorial distinction here is that ZF omits Choice.

Concept sources

Wikipedia — Zermelo–Fraenkel set theory — Zermelo–Fraenkel set theory · encyclopedic reference · source ID wp-zf
nLab — ZF — ZF · mathematical reference · source ID nlab-zf
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory
Stanford Encyclopedia of Philosophy — Set Theory — Set Theory · scholarly reference · source ID sep-set-theory

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

First-order theory Zermelo–Fraenkel set theory (ZF)

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: first_order_theory_to_zf
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: ZF theory

Authored explanation

ZF is a first-order theory in the language with membership as its only nonlogical relation.

How to interpret this relation type

The target is a member or subtype of the broader source class.

Relation sources

Wikipedia — Zermelo–Fraenkel set theory — Zermelo–Fraenkel set theory · encyclopedic reference · source ID wp-zf
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.