you can’t talk about the integers or the reals in first-order logic.
It’s more accurate to say that you can’t talk about arbitrary subsets of the integers or the reals in first-order logic.
Accepting that the integers have a supremum is a whole different kettle of fish from accepting that the collection of finite integers seems to go on without bound.
I agree. This is the difference between completed and potential infinity. Nelson.
Second, taking a supremum once, using a clearly defined computable notation and a halting machine that can compare any two representations, is a whole different kettle of fish than talking about the supremum of all possible ways to define countable well-orderings to and beyond computable recursion.
I’m not so sure. Everything you’ve ever talked about, uncountable ordinals and all, you’ve talked about using computable notation. Computable, period is a whole different kettle of fish.
It’s more accurate to say that you can’t talk about arbitrary subsets of the integers or the reals in first-order logic.
I agree. This is the difference between completed and potential infinity. Nelson.
I’m not so sure. Everything you’ve ever talked about, uncountable ordinals and all, you’ve talked about using computable notation. Computable, period is a whole different kettle of fish.