Harry Potter and the Methods of Rationality discussion thread, part 8

Unnamed

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Eliezer Yudkowsky which is not supposed to be publicly available (which includes public statements by Eliezer that have been retracted).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.

Update: Discussion has moved on to a new thread.

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Eliezer Yudkowsky which is not supposed to be publicly available (which includes public statements by Eliezer that have been retracted).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.

I would be interested in knowing if there is any second-order system which is strong enough to talk about continuity, but not to prove the existence of a first uncountable ordinal.

The J_2 referenced in this subthread shoud do the trick. (See in particular number 6 in the page linked to by shinoteki there).

4[anonymous]15y

van den Dries, "Tame topology and o-minimal structures," Cambridge U Press 1998 develops a lot of 20th century geometry in a first order theory of real numbers. You can do enough differential geometry in this setting to do e.g. general relativity.

5JoshuaZ15y

I can do better. I can give you a complete, decidable, axiomatized system that does that: first order real arithmetic. However, in this system you can't talk about integers in any useful way. We can do better than that: first order real arithmetic + PA + a set of axioms embedding the PA integers into R in the obvious way. This is a second order system where I can't talk about uncountable ordinals. However, this system doesn't let us talk about sets. Note that in both these cases we've done this by minimizing how much we can talk about sets. Is there some easy way to do this where we can talk about set a reasonable amount? I'm not sure. Answering that may be difficult (I don't think the question is necessarily well-defined.) However, I suspect that the following meets one's intuition as an affirmative answer: Take ZFC without regularity, replacement or infinity, choice, power set or foundation. Then add as an axiom that there exists a set R that has the structure of a totally ordered field with the least-upper bound property. This structure allows me to talk about most things I want to do with the reals while probably not being able to prove nice claims about Hartogs numbers which should make proving the existence of uncountable ordinals tough. It would not surprise me too much if one could get away with this system with the axiom of the power set thrown also. But it also wouldn't surprise me either if one can find sneaky ways to get info about ordinals. Note that none of these systems are at all natural in any intuitive sense. With the exception of first-order reals they are clear attempts to deliberately lobotomize systems. (ETA: Even first order reals is a system which we care about more for logic and model theoretic considerations than any concrete natural appreciation of the system.) Without having your goal in advance or some similar goal I don't think anyone would ever think about these systems unless they were a near immortal who was passing the time by

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Harry Potter and the Methods of Rationality discussion thread, part 8

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