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.