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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)?

Many thanks for your help!

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marked as duplicate by Asaf Karagila logic Jan 22 '18 at 5:17

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  • $\begingroup$ Construct two models, where all the other axioms hold. In model #1, the axiom of interest does hold, in model #2, it doesn't. $\endgroup$ – Lee Mosher Jan 22 '18 at 3:02
  • $\begingroup$ I would think to prove the axiom only using itself? $\endgroup$ – Badr B Jan 22 '18 at 3:05
  • $\begingroup$ I would probably add a handful of other duplicates once I am on a proper keyboard. $\endgroup$ – Asaf Karagila Jan 22 '18 at 5:18
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Think of it as an argument 'If all the other axioms hold, then this axiom holds', and produce a counterexample, i.e. find a scenario where all the other axioms hold but the one axiom does not.

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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 $\Gamma$.

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