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Sewing-Machine comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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The existence of the real number line is one thing. The existence of an uncountable ordinal is another. When you consider the hierarchies of uncomputable ordinals to their various Turing degrees that are numbered among the countable ordinals, and that which countable ordinals you can constructively well-order strongly corresponds to the strength of your proof theory and which Turing machines you believe to halt, and when you combine this with the Burali-Forti paradox saying that the predicate "well-ordered" cannot be self-applicable, even though any given collection of well-orderings can be well-ordered...
...I just have trouble believing that there's actually any such thing as an uncountable ordinal out there, because it implies an absolute well-ordering of all the countable well-orderings; it seems to have a superlogical character to it.
For essentially the same reasons I have trouble believing that the first infinite ordinal exists.
Finite ordinals are computable, but otherwise your remarks still apply if you swap out "countable" for "finite." According to ZF there are uncomputable sets of finite ordinals, so you can't verify that they are well-ordered algorithmically.
So what you're saying is that you don't believe the natural numbers exist.
The natural numbers exist in about the strongest possible sense: I can get a computer program to spit them out one by one, and it won't stop until it runs out of resources. It's more accurate to say I don't believe that they're well-ordered, see here.
You might find my reasoning preposterous, I only wanted to point out that it's essentially the same as EYs reasoning about uncountable ordinals.