# How to prove that a axiom in a certain system is independent of the other ones? [duplicate]

As I'm learning Zermelo–Fraenkel set theory, a question arises: How do we know (or prove) a axiom in a axiomatic system is independent of the other ones? (that means we can not prove one from the rest)?

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You can prove that a formula $\phi$ is independent of a set of formulas $\Gamma$, if you can obtain two structures $\mathcal{M,M'}$ such that $$\mathcal{M}\models\Gamma\cup\phi \ \ \text{and} \ \mathcal{M'}\models\Gamma\cup\neg\phi$$ Whit this and completeness theorem you can conclude that $\phi$ not follows of $\Gamma$.