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Eliezer Yudkowsky which is not supposed to be publicly available (which includes public statements by Eliezer that have been retracted).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.
Yes, ZFC is quite enough to imply the existence of the first uncountable ordinal.
On the other hand, I don't see what's unbelievable about such a thing; it's just (the order type of) the set of all countable ordinals, and I don't see why it's unbelievable that there is such a set. (That is, if you're going to accept uncountable sets in the first place; and if you don't want that, then you can criticise ZFC on far more basic grounds than anything about ordinals.)
I think the fact that considering the set of all ordinals leads to trouble should make you somewhat uncomfortable with the set of countable ordinals.
I'd go a step further and say you should be uncomfortable with the set of finite ordinals. But maybe these are the more basic criticisms you're talking about.