What types of axioms retain consistency of ZFC under additional axioms about its consistency?

Question

My question is pretty simple. If ZFC does not prove that it is not consistent, then can we add the axiom to ZFC that it proves it is not consistent and is consistent and achieve equiconsistency? I think I got confused because obviously we can add the axiom to ZFC that it is not consistent and achieve equiconsistency, but can we also do the same when we say ZFC proves it is not inconsistent (while it does not) and that it is consistent? This was inspired by the list of options of consistency of ZFC, where they say ZFC could be consistent but prove it is not. To clarify, as the comments have pointed out, I want to know if ZFC + ZFC proves not(Con(ZFC)) + Con(ZFC) is consistent if ZFC does not prove not(Con(ZFC)).

Answer 1

As you say in the comments, ZFC not proving $\lnot$Con(ZFC) means the same thing as ZFC + Con(ZFC) being consistent. Likewise, “ZFC proves $\lnot$Con(ZFC)” means the same thing as “ZFC + Con(ZFC) is inconsistent” (and ZFC—or for that matter pretty much any reasonable theory—can prove this equivalence). So you are asking if the statement “ZFC + Con(ZFC) is inconsistent” is consistent with ZFC + Con(ZFC). The answer is yes, by the second incompleteness theorem applied to ZFC + Con(ZFC).