4 ms·
Do the incompleteness theorems not apply here?
by stogot 3y ago
Do the incompleteness theorems not apply here?
- lmm 3y agoThey start with a large cardinal assumption which is the usual workaround - you have a proof that this theory is consistent if a large cardinal like that exists, which is something you can't prove within the theory.
- randallholmes 3y agoIt'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.
- randallholmes 3y agoand 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.
- lmm 3y agoHow 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 3y agoYou 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.
- lmm 3y ago> Godel's theorem shows that that context has to be stronger than T. This isn't some kind of obstruction, it is reality. What distinction are you drawing? I see no contradiction between something being reality and something being an obstruction to be worked around.
- randallholmes 3y agoI 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.