---
tag: WP6KSK5ZD9
kind: concept
name: Standard model (set theory)
aliases: [countable-transitive-model, set-model, standard-transitive-model, transitive-model]
sources:
  - wikipedia: en:18391755@r1367617467
    locator: Standard model (set theory)
    note: adopted
    mention: wikipedia://en/18391755
    sections: [Summary]
same_as:
  wikidata: Q117715071
  wikipedia: Standard model (set theory)
inventory: [5KGJHE89G5]
---

## Summary

In set theory, a **standard model** for a theory $T$ (in the language of set theory) is a model $M$ for $T$ where the membership relation $\in_{M}$ is the same as the membership relation $\in$ of a set theoretical universe $V$ (restricted to the domain of $M$). In other words, $M$ is a substructure of $V$. A standard model $M$ that satisfies the additional transitivity condition that $x \in y \in M$ implies $x \in M$ is a **standard transitive model** (or simply a **transitive model**).