Sewing-Machine comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong

8 Post author: Unnamed 25 August 2011 02:17AM

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Comment author: [deleted] 08 September 2011 03:34:40AM 2 points [-]

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.

I have my problems with the other two, but this is the only one I don't understand. What do you mean?

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

You seem to accept the notion that all finite numbers have a supremum. Why not just iterate whatever process accounts for that?

Comment author: Eliezer_Yudkowsky 08 September 2011 04:07:34AM 0 points [-]

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

You seem to accept the notion that all finite numbers have a supremum. Why not just iterate whatever process accounts for that?

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.

Comment author: [deleted] 08 September 2011 01:15:43PM 4 points [-]

you can't talk about the integers or the reals in first-order logic.

It's more accurate to say that you can't talk about arbitrary subsets of the integers or the reals in first-order logic.

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.

I agree. This is the difference between completed and potential infinity. Nelson.

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.

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.

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