I am pretty sure the is not obstacle for applying the successor function to the infinite set. And then there is the construction mirroring ω + ω. If you have the infinite set and it has many successors what limits one to not do the inductive set trick again to this situation?
I kinda know that if you assume a special inductive set that is only one “permitted application” of it and a “second application” would need a separate ad hoc axiom.
Then if we have “full blown” transfinite recursion we just allow that second-level application.
New assumtions assume that there are old assumptions. If we just have non-proof “I have feeling it should be that way” we have a pre-axiomatic system before hand. If we don’t aim to get the same theorems then “minimal change to keep intact” doesn’t make sense. The conneciton here is whether some numbers “fakely exist” where a fake existence could be that some axiom says the thing exist but there is no proof/construction that results in it. A similar kind of stance could be that real numbers are just a fake way to talk about natural numbers and their relations. One could for example note that reals are innumerable but proofs are discrete so almost all reals are undefineable. If most reals are undefinable then unconstructibility by itself doesn’t make transfinites any less real. But if the real field can establish some kind of properness then the same avenues of properness open up to make transfinites “legit”.
I am not that familar how limits connect to the foundamentals but if that route-map checks out then transfinites should not be any ickier than limits.
I am pretty sure the is not obstacle for applying the successor function to the infinite set. And then there is the construction mirroring ω + ω. If you have the infinite set and it has many successors what limits one to not do the inductive set trick again to this situation?
I kinda know that if you assume a special inductive set that is only one “permitted application” of it and a “second application” would need a separate ad hoc axiom.
Then if we have “full blown” transfinite recursion we just allow that second-level application.
New assumtions assume that there are old assumptions. If we just have non-proof “I have feeling it should be that way” we have a pre-axiomatic system before hand. If we don’t aim to get the same theorems then “minimal change to keep intact” doesn’t make sense. The conneciton here is whether some numbers “fakely exist” where a fake existence could be that some axiom says the thing exist but there is no proof/construction that results in it. A similar kind of stance could be that real numbers are just a fake way to talk about natural numbers and their relations. One could for example note that reals are innumerable but proofs are discrete so almost all reals are undefineable. If most reals are undefinable then unconstructibility by itself doesn’t make transfinites any less real. But if the real field can establish some kind of properness then the same avenues of properness open up to make transfinites “legit”.
I am not that familar how limits connect to the foundamentals but if that route-map checks out then transfinites should not be any ickier than limits.