---
tag: QKCAEG2Y0E
kind: statement
role: proposition
name: Axiom independence
aliases: [independent-axiom]
sources:
  - wikipedia: en:6118589@r1358356173
    locator: Axiom independence
    note: adopted
    mention: wikipedia://en/6118589
    sections: [Statement]
same_as:
  wikidata: Q4830547
  wikipedia: Axiom independence
inventory: [1JM6WZKD3B]
---

## Statement

An axiom $P$ is **independent** of a theory $T$, i.e., a collection of sentences, if and only if neither $P$ nor its negation $\neg P$ is provable from $T$. Equivalently, $P$ and $\neg P$ are each consistent with $T$, i.e., some models of $T$ make $P$ true and others make $\neg P$ true. Proving the independence of $P$ from $T$ is usually done by demonstrating the existence of such models.