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?
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...
Update: Discussion has moved on to a new thread.
The hiatus is over with today's publication of chapter 73, and the previous thread is approaching the 500-comment threshold, so let's start a new Harry Potter and the Methods of Rationality discussion thread. This is the place to discuss Eliezer Yudkowsky's Harry Potter fanfic and anything related to it.
The first 5 discussion threads are on the main page under the harry_potter tag. Threads 6 and on (including this one) are in the discussion section using its separate tag system. Also: one, two, three, four, five, six, seven. The fanfiction.net author page is the central location for information about updates and links to HPMOR-related goodies, and AdeleneDawner has kept an archive of Author's Notes.
As a reminder, it's often useful to start your comment by indicating which chapter you are commenting on.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: