But it really is mind-bending: if you, instead, add a theorem to the system (say, ZFC) which states, "ZFC is consistent", this system (ZFC+Con(ZFC)) is consiste

But it really is mind-bending: if you, instead, add a theorem to the system (say, ZFC) which states, "ZFC is consistent", this system (ZFC+Con(ZFC)) is consistent! Even more strangely, if you instead add "ZFC is not consistent", this system (ZFC+notCon(ZFC)) is also consistent. We call these "self-hating theories."