Note that without replacement, you can't construct the von Neumann ordinal omega*2, or any higher ones, so certainly not omega_1. Of course, this doesn't prevent uncountable well-ordered sets (obviously these follow from choice, though I guess you're taking that out as well), but you need replacement to show that every well-ordered set is isomorphic to a von Neumann ordinal.
So I don't think that this should prevent the construction of an order of type omega_1, even if it can't be realized as a von Neumann ordinal. Of course losing canonical representatives means you have to talk about equivalence classes, but if all we want to do is talk about omega_1, it suffices to consider well-orderings of subsets of N, so that the equivalence classes in question will in fact be sets. Maybe there's some other technical obstacle I'm missing here (like it somehow wouldn't be the first uncountable ordinal despite being the right order?) -- this isn't really my area and I haven't bothered to work through it, I can try that later -- but I wouldn't expect one.
Maybe there's some other technical obstacle I'm missing here
There's not. The Hartog's number construction gives us the set H(N) of all isomorphism classes of well-orders on subsets of any fixed countably infinite set, and we can prove that H(N) is uncountable and every proper initial segment of H(N) is countable, using power set and separation (but only bounded separation) but not replacement. I verified this just now by looking at Wikipedia's article on Hartog's number and checking through the proof myself.
The next step (step 4 in Wikipedia, ETA: whi...
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: