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I have trouble understanding this wiki page, concerning inaccessible cardinals.

First "These statements are strong enough to imply the consistency of ZFC". I guess what they mean by that is if $\kappa$ is a strongly inaccessible cardinal, then $V_\kappa$ is a model of ZFC, where $V$ is the Von Neumann hierarchy.

Then they argue that a proof in ZFC, assuming ZFC is consistent, of the consistency of ZFC+$\kappa$ is impossible. Because it would prove the consistency of ZFC, violating Gödel's incompleteness theorem.

I don't understand this, I imagine a relative consistency proof would rather be a model transformation : given a model of ZFC as input, produce a model of ZFC+$\kappa$ as output. Then we could continue producing $V_\kappa$ to get another model of ZFC. Where is the contradiction in that ? To me it seems we first assumed a model of ZFC, then produced another one. ZFC didn't produce a model of itself from scratch.

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Apply the incompleteness theorem to $ZFC+\exists \text{an inaccessible} $

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  • $\begingroup$ So simple... sorry I didn't see that myself $\endgroup$
    – V. Semeria
    Aug 4 '18 at 15:58

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