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Eliezer_Yudkowsky comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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http://en.wikipedia.org/wiki/Second-order_logic#Expressive_power - you can't talk about the integers or the reals in first-order logic. You can have first-order theories with the integers as a model, but they'll have models of all other cardinalities too. http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem
First of all, I've never seen an aleph-null, just one, two, three, etc. 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. 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.
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.