Dropping down to Peano Arithmetic for a moment. We can consider adding a new constant 'c' for a natural number to the language along with the following infinit

Dropping down to Peano Arithmetic for a moment. We can consider adding a new constant 'c' for a natural number to the language along with the following infinite list of axioms about this remarkable constant: - 0 < c - 1 < c - 2 < c - 3 < c ... Adding all these axioms is consistent. I.e. you can do induction upto 'c', whatever it is. Why is it consistent? Because if there was a contradiction, the proof of such a contradiction would be finite, and hence can only use a finite number of these new axioms (this is a so-call compactness argument). But clearly any finite subset of this list of axioms is consistent because it has a model where c is just defined to be 1 more than the largest numeral appearing in that list. But all those infinite number of axioms taken together creates an unsound system because it claims that 'c' denotes a natural number that is larger than every written numeral. Heading back to ZFC land, it turns out that (assuming ¬Con(ZFC) is independent of ZFC) adding ¬Con(ZFC) to ZFC similarly is similarly unsound in that it yields only models that have elements that are larger than every written numeral.