I don’t believe it’s good math until it becomes possible to talk about the first uncountable ordinal, in the way that you can talk about the integers. Any first-order theory of the integers, like first-order PA, will have some models containing supernatural numbers, but there are many different sorts of models of supernatural numbers, you couldn’t talk about the supernaturals the way you can talk about 3 or the natural numbers. My skepticism about “the first uncountable ordinal” is that there would not exist any canonicalizable mathematical object—nothing you could ever pin down uniquely—that would ever contain the first uncountable ordinal inside it, because of the indefinitely extensible character of well-ordering. This is a sort of skepticism of Platonic existence—when that which you thought you wanted to talk about can never be pinned down even in second-order logic, nor in any other language which does not permit of paradox.
You seem to keep forgetting that the whole notion of “second-order logic” does not make sense without some ambient set theory. (Unless I am greatly misunderstanding how second-order logic works?) And if you have that, then you can pin down the natural numbers (and the first uncountable ordinal) in first-order terms in this larger theory.
Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense. It’s just that the ambient models in the second-order theory include collections of possible predicates of any objects that get predicates attached, or if you prefer, people who speak in second-order logic think that it makes as much sense to say “all possible collections that include some objects and exclude others, but still include and exclude only individual objects” as “all objects”.
Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense.
Well, it makes sense to me without any models. I can compute, prove theorems, verify proofs of theorems and so on happily without ever producing a “model” for the natural numbers in toto, whatever that could mean.
Okay… I now know what an ordinal number actually is. And I’m trying to make more sense out of your comment...
So, re-reading this:
or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness, is something we have rather less evidence for
So if I understand you correctly, you don’t trust anything that can’t be defined up to isomorphism in second-order logic, and “the set of all countable ordinals” is one of those things?
(I never learned second order logic in college...)
In other words, a first uncountable ordinal may be perfectly good math, but it’s not physics?
I don’t believe it’s good math until it becomes possible to talk about the first uncountable ordinal, in the way that you can talk about the integers. Any first-order theory of the integers, like first-order PA, will have some models containing supernatural numbers, but there are many different sorts of models of supernatural numbers, you couldn’t talk about the supernaturals the way you can talk about 3 or the natural numbers. My skepticism about “the first uncountable ordinal” is that there would not exist any canonicalizable mathematical object—nothing you could ever pin down uniquely—that would ever contain the first uncountable ordinal inside it, because of the indefinitely extensible character of well-ordering. This is a sort of skepticism of Platonic existence—when that which you thought you wanted to talk about can never be pinned down even in second-order logic, nor in any other language which does not permit of paradox.
You seem to keep forgetting that the whole notion of “second-order logic” does not make sense without some ambient set theory. (Unless I am greatly misunderstanding how second-order logic works?) And if you have that, then you can pin down the natural numbers (and the first uncountable ordinal) in first-order terms in this larger theory.
Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense. It’s just that the ambient models in the second-order theory include collections of possible predicates of any objects that get predicates attached, or if you prefer, people who speak in second-order logic think that it makes as much sense to say “all possible collections that include some objects and exclude others, but still include and exclude only individual objects” as “all objects”.
Well, it makes sense to me without any models. I can compute, prove theorems, verify proofs of theorems and so on happily without ever producing a “model” for the natural numbers in toto, whatever that could mean.
Hmmm…
::goes and learns some more math from Wikipedia::
Okay… I now know what an ordinal number actually is. And I’m trying to make more sense out of your comment...
So, re-reading this:
So if I understand you correctly, you don’t trust anything that can’t be defined up to isomorphism in second-order logic, and “the set of all countable ordinals” is one of those things?
(I never learned second order logic in college...)