I was reading the wikipedia article of Gödel's incompleteness theorems and I saw that it has been proven that ZFC + "there exists an inaccessible cardinal" (which I'll call "ZFC + IC") proves that ZFC is consistent.
What would happen if we took ZFC + $\neg$IC and proved that ZFC is consistent with it?
Would that mean that ZFC is consistent in general? Would we also have to prove that both ZFC + IC and ZFC + $\neg$IC are consistent?
I don't know much about set theory and cardinals, so I would appreciate it if the answer doesn't involve too many fancy terms
That would be a proof that ZFC shows that ZFC is consistent. Godel showed that there is no such proof. (Unless of course ZFC is in fact inconsistent, in which case all these proofs exist, with all but the first one trivial...)