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:
You do not need to rot13 anything about HP:MoR or the original Harry Potter series unless you are posting insider information from Eliezer Yudkowsky which is not supposed to be publicly available (which includes public statements by Eliezer that have been retracted).
If there is evidence for X in MOR and/or canon then it's fine to post about X without rot13, even if you also have heard privately from Eliezer that X is true. But you should not post that "Eliezer said X is true" unless you use rot13.
OK, you say you don't accept that sort of uncomputable leap to the end. The problem is that, AIUI, you're already accepting it as soon as you accept the power set of N. (Of the various "axioms of power" of ZFC, power set is the only one needed here. And if you just want omega_1, you don't need arbitrary power sets, just that of N. I mean really you want P(N x N), but since N is in an easily-described bijection with N x N, it shouldn't make a difference; just use a pairing function instead of proper ordered pairs.) The construction of omega_1 from P(N) is pretty straightforward, really, and doesn't use any of ZFC's other powerful axioms. Maybe you can somehow have the reals without P(N)? I.e. without binary expansions? shrug This is getting rather far away from what I know. Constructivists -- well, not the milder ones who just reject excluded middle, but the stricter ones who don't like impredicativity (whatever that might be, don't ask me) -- don't accept the axiom of power sets; they consider it just as much an unjustified leap to the end.
Of course you could always try summoning TobyBartels and ask him how the constructivists do it. When you say these sorts of things I'm a little of surprised you haven't gone constructivist already. But I guess you like classical logic. :)
(By the "axioms of power", I mean replacement, power set, and choice; the ones anyone might object to. Well, foundation is objectionable too, but it's more of an axiom of weakness. Healing Salve as opposed to Ancestral Recall. :P Also looking things up apparently the no-impredicativity constructivists insist on weakening axiom of separation as well? Well, I think their weaker version should suffice here. Again, I am saying these things without carefully checking them because hopefully TobyBartels will show up and correct me if I am wrong. :) )
You either need P(P(N)) or something like an axiom of quotient sets to take the equivalence classes that are the actual elements of this version of omega_1. I presume (but haven't checked) that this is why J_2 has R but not omega_1 (although J_2 is not written in set-theoretic language, so you have to encode these).
Assuming you accept classical logic, then P(N) may be construct... (read more)