[randallholmes]

It's not really a workaround. Whenever we are proving the consistency of a theory T, we are implicitly working in a stronger system. That is just how consistency proofs are done. The incompleteness theorems do not say that we cannot prove the consistency of theories, or build models of theories...in a generally accepted theory such as ZFC. They do say that if we want to prove the consistency of ZFC itself we would need to work in an even stronger system.

[lmm]

How is that not a workaround? It's "just how consistency proofs are done" because it's a standard, popular workaround for the problem. We absolutely would prove consistency of ZFC and similar theories in ZFC if we could, we only work in stronger systems because we have to.

[randallholmes]

You don't work around what is impossible. A consistency proof for a theory T is usually a construction of a model of that theory in some context we have confidence in. Godel's theorem shows that that context has to be stronger than T. This isn't some kind of obstruction, it is reality. And there are strong systems we have confidence in.

[randallholmes]

I have the same objection when people talk about defining set theories in such a way as to avoid the paradoxes. We don't avoid or work around the paradoxes: they are mistakes. We simply do things correctly, we do what we can do.

[randallholmes]

and we really don't use a large cardinal assumption...the existence of beth_omega_1 is really small potatoes. But, it is stronger than NF.