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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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I think we may have something of a clash of backgrounds here. The reason I'm inclined to take the real continuum seriously is that there are numerous physical quantities that seem to be made of real or complex numbers. The reason I take mathematical induction seriously is that it looks like you might always be able to add one minute to the total number of minutes passed. The reason I take second-order logic seriously is that it lets me pin down a single mathematical referent that I'm comparing to the realities of space and time.
The reason I'm not inclined to take the least uncountable ordinal seriously is because, occupying as it does a position above the Church-Kleene ordinal and all possible hypercomputational generalizations thereof, it feels like talking about the collection of all collections - the supremum of an indefinitely extensible quality that shouldn't have a supremum any more than I could talk about a mathematical object that is the supremum of all the models a first-order set theory can have. If set theory makes the apparent continuum from physics collide with this first uncountable ordinal, my inclination is to distrust set theory.
I have my problems with the other two, but this is the only one I don't understand. What do you mean?
You seem to accept the notion that all finite numbers have a supremum. Why not just iterate whatever process accounts for that?
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.