Nope they're perfectly allowed! ZFC + ~Con(ZFC) is a perfectly valid axiom system that is consistent if and only if ZFC is consistent. In fact as a result ZFC

Nope they're perfectly allowed! ZFC + ~Con(ZFC) is a perfectly valid axiom system that is consistent if and only if ZFC is consistent. In fact as a result ZFC + ~Con(ZFC) cannot prove any theorem that would contradict ZFC. Indeed basically all of modern mathematics looks the same under ZFC + ~Con(ZFC) as it does under ZFC. But how can that be? What happened to the Principle of Explosion? How is this even coherent? The trick is that ~Con(ZFC) being true doesn't mean that... well it's true (in the usual way that we talk about it). This is why I find the discussion of truth in Godel's incompleteness theorems to be distractions without a backdrop of model theory. Here it is important to delineate between an "internal" and "external" perspective. For a formal system S, the statement Con(S) is at the end of the day an arithmetic statement postulating the existence of a natural number satisfying certain constraints (this is why Godel's incompleteness theorems require the ability to perform arithmetic). This is the internal perspective, asking about a natural number. Externally, we know of the correspondence between the natural numbers and proofs and sentences, but that is external to our theory. ZFC cannot use the statement ~Con(ZFC), which we externally know encodes a proof that 0 = 1, to prove internally that 0 in fact is equal to 1. Thus we avoid the Principle of Explosion within ZFC. Indeed in this case, it turns out that what ZFC + ~Con(ZFC) thinks is a natural number is not unique! Different universes of sets that agree on ZFC might have different notions of natural numbers. Indeed the element that ZFC + ~Con(ZFC) thinks is a natural number satisfying ~Con(ZFC) might not be something another version of ZFC thinks is a natural number. And so when externally we go to try to prove that ~Con(ZFC) shows that ZFC is indeed inconsistent our mapping breaks down because we disagree that the witness to ~Con(ZFC) is a natural number externally. This whole pile of stuff is why I think using the word "truth" hurts more than helps unless everyone has a model theory background.