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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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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.