Okay, I kinda understood where I am wrong spiritually-intuitively, but now I don’t understand where I’m wrong formally. Like which inference in chain
not Consistent(ZFC) → some subsets of ZFC don’t have a model → some subsets of ZFC + not Consistent(ZFC) don’t have a model → not Consistent(ZFC + not Consistent(ZFC))
It’s completely valid. And we can simplify it further to:
not Consistent(ZFC) → not Consistent(ZFC + not Consistent(ZFC))
because if a set of axioms is already inconsistent, then it’s inconsistent with anything added. But you still won’t be able to actually derive a contradiction from this.
Okay, I kinda understood where I am wrong spiritually-intuitively, but now I don’t understand where I’m wrong formally. Like which inference in chain
not Consistent(ZFC) → some subsets of ZFC don’t have a model → some subsets of ZFC + not Consistent(ZFC) don’t have a model → not Consistent(ZFC + not Consistent(ZFC))
is actually invalid?
It’s completely valid. And we can simplify it further to:
not Consistent(ZFC) → not Consistent(ZFC + not Consistent(ZFC))
because if a set of axioms is already inconsistent, then it’s inconsistent with anything added. But you still won’t be able to actually derive a contradiction from this.
Edit: I think the right thing to do here is look at models for PA + not consistent(PA). I can’t find a nice treatment of this at the moment, but here’s a possibly wrong one by someone who was learning the subject at the time: https://angyansheng.github.io/blog/a-theory-that-proves-its-own-inconsistency