= 27d567bc2d48d9f40e9ccc20ea370b15)
Cool.
For those who don't want to wait until April 7th, Udacity is scheduled to launch their own Intro to Data Science on Feb 5th (this Wednesday). I expect it to be easier/shallower than the Hopkins/Coursera sequence, but it wins on actionability.
I meant that (conjecturally) for every measure, there exists a cardinal kappa such that mu({M: |M| > kappa}) = 0. Anyway, I guess as you've demonstrated the set/class thing isn't a big problem, but it is something to watch out for.
Okay, that makes sense.
No, I was observing the following: mu is countably additive, and the set of theories is countable. Hence the measure of the total space is the sum of the measures of the theories, so the measures of the theories must sum to 1. Now it's clear that at every step i of the process, the sum of the measures of the (incomplete) theories so obtained is 1. But it's not immediately clear to me that this holds in the limit.
However, I just realized my mistake, which is that the set of theories isn't always countable (there are countably many sentences, but a theory is a subset of the sentences; for instance, consider the language with countably many unary relations and a constant symbol). In particular, I believe it's countable if and only if the sum is preserved in the limit, so we're fine.
For a countable language L and theory T (say, with no finite models), I believe the standard interpretation of "space of all models" is "space of all models with the natural numbers as the underlying set". The latter is a set with cardinality continuum (it clearly can't be larger, but it also can't be smaller, as any non-identity permutation of the naturals gives a non-identity isomorphism between different models).
Moreover this space of models has a natural topology, with basic open sets {M: M models phi} for L-sentences phi, so it makes sense to talk about (Borel) probability measures on this space, and the measures of such sets. (I believe this topology is Polish, actually making the space Borel isomorphic to the real numbers.)
Note that by Lowenheim-Skolem, any model of T admits a countable elementary substructure, so to the extent that we only care about models up to some reasonable equivalence, countable models (hence ones isomorphic to models on the naturals) are enough to capture the relevant behavior. (In particular, as pengvado points out, the Christiano et al paper only really cares about the complete theories realized by models, so models on the naturals suffice.)
In response to
Rewiring my Brain: (gentle) Help Appreciated
There seems to be a mismatch between your description of the problem and your description of the solution. If you are already able to grasp concepts, but not to apply them rapidly, then the solution ought to be on the application side, not on the comprehension side as implied by "turbocharge [...] ability to absorb and deeply comprehend".
This being so, I suggest you practice doing problems. Lots of them. Go through your physics textbook (or calculus, or whatever) problem by problem, and do them all. Practice makes perfect.
In addition, it is my experience that people who say they grasp the concepts but can't apply them haven't actually grasped the concepts at all. I have yet to encounter anyone (including myself) who really fundamentally gets it (where 'it' may be physics, calculus, or even algebra) without doing a lot of dang problems.
Finally: It may be the case that your current learning effort is a bit misdirected. You say you want a career in a software-ish direction; you don't need to know a whole lot of either math or physics for that. You mention pre-requisites, so perhaps it is unavoidable, but any additional effort besides coursework may be better spent on programming. Which, again, you should learn by doing exercises, not reading books. Fire up emacs, write "Hello World", compile from the command line, run ditto. Iterate from there. For the sake of the absent gods do not start with an IDE or anything that hides the compilation step.
For the sake of the absent gods do not start with an IDE or anything that hides the compilation step.
Could you say more about why this is important for beginning programmers?
It's from a sermon in which Wesley advocates that Christian should "gain all you can", "save all you can", and "give all you can" — a teaching somewhat similar to efficient altruism.
doesn't look very similar to me - it's missing the "efficient" part; focusing on "how much do you give?" instead of "is it doing any fricking good?"
While you're certainly technically correct, it's an easy/common mistake for people to focus on the "save all you can" part, overlooking "gain all you can" opportunities. The EA movement is notable for proactively trying to counter this mistake, and apparently so is John Wesley.
I suppose the question is: What should you do if you're offered a bet on whether the dart will hit the target or not?
There's no way to avoid the question other than arguing somehow that you'll never encounter an immeasurable set.
In response to
Open Thread, September 30 - October 6, 2013
If I make a target, but instead of making it a circle, I make it an immeasurable set, and you throw a dart at it, what's the probability of hitting the target?
In other words, "what is the measure of an unmeasurable set?". The question is wrong.
In response to
Mistakes repository
Several popular comments say something to the effect of "I was too arrogant to just get with the program and cooperate with the other humans".
The biggest of my own arrogant mistakes was not taking CS/programming very seriously while in college because I was dead set on becoming a mathematician, and writing code was "boring" compared to math. Further arrogance: I wasn't phased by the disparity between the number of graduating Ph. D.'s and the number of academic jobs.
I found out in grad school that my level of talent in mathematics, while somewhat rare, was certainly not so exceedingly rare that real world considerations would not apply to me.
I've since changed my attitude, and I'm working on fixing this mistake.
In response to
MIRI course list book reviews, part 1: Gödel, Escher, Bach
GEB will take you from superficial knowledge to full grok.
A word of caution: there is a risk when reading popular science/math books like GEB of coming away feeling like one understands something at a higher level than one actually does, particularly if one hasn't already studied the subject formally.
If one has formally studied incompleteness before, it's easy to wave away standard primitive recursive derivations (e.g. the proof predicate) as tedious and trivial and beside the main point, but having this attitude the first time around could be dangerous.
I read GEB years ago, and recall liking it quite a bit, though I disagree with Louie Helm's endorsement of GEB as a course reference, at least not without supplementation from a "standard" source like these notes (or any formal logic textbook).
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Because the length of Scott's Moloch post greatly exceeds my working memory (to the extent that I had trouble remembering what the point was by the end) I made these notes. I hope this is the right place to share them.
Notes on Moloch (ancient god of child sacrifice)
http://slatestarcodex.com/2014/07/30/meditations-on-moloch/
Intro - no real content.
Moloch as coordination failure: everyone makes a sacrifice to optimize for a zero-sum competition, ends up with the same relative status, but worse absolute status.
Existing systems are created by incentive structures, not agents, e.g. Las Vegas caused by a known bias in human reward circuitry, not optimization for human values.
But sometimes we move uphill anyway. Possible explanations:
Technology/ingenuity creates new opportunities to fall into such traps. Technology overcomes physical limitations, consumes excess resources. Automation further decouples economic activity from human values. Technology can improve coordination, but can also exacerbate existing conflicts by giving all sides more power.
AGI opens up whole new worlds of traps: Yudkowsky's paperclipper, Hanson's subsistence-level ems, Bostrom's Disneyland with no children.
6 & 7. Gnon - basically the god of the conservative scarcity mindset. Nick Land advocates compliance; Nyan wants to capture Gnon and build a walled garden. Scott warns that Moloch is far more terrifying than Gnon and will kill both of them anyway.
8 & 9. So we have to kill this Moloch guy, by lifting a better God to Heaven (Elua).