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Eliezer Yudkowsky which is not supposed to be publicly available (which includes public statements by Eliezer that have been retracted).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.
One thing bothering me -- is there any way to define a well-founded set without using infinitary reasoning? It's easy enough to say that all sets are well-founded without it, by just stating that ∈ is well-founded -- I mean, that's what the standard axiom of foundation does, though with the classical definition -- but in contexts where that doesn't hold, you need to be able to distinguish a well-founded set from an ill-founded one. Obvious thing to do would be to take the transitive closure of the set and ask if ∈ is well-founded on that, but what bugs me is that constructing the transitive closure requires infinitary reasoning as well. Is there something I'm missing here?
I know one way; it cannot be stated in ZFC↺ (ZFC without foundation), but it can be stated in MK↺ (the Morse–Kelley class theory version): a set is well-founded iff it belongs to every transitive class of sets (that is every class K such that x ∈ K whenever x ⊆ K); it is immediate that we may prove properties of these sets by induction on membership, and a set is well-founded if all of its elements are, so this is a correct definition. However, it requires quantification over all classes (not just sets) to state.