I think “equiconsistent” just means that (one can prove that) if one is consistent, then the other is also consistent. Doesn’t mean the collections of things t

I think “equiconsistent” just means that (one can prove that) if one is consistent, then the other is also consistent. Doesn’t mean the collections of things they can prove can’t differ. For a weak example, aiui, a formal system is equiconsistent with any conservative extension of it, but the conservative extension can (in the cases one would usually speak of) prove some statements that aren’t in the language of the original system.