Qiaochu_Yuan comments on Godel's Completeness and Incompleteness Theorems - Less Wrong

34 Post author: Eliezer_Yudkowsky 25 December 2012 01:16AM

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Comment author: Qiaochu_Yuan 29 December 2012 09:16:37PM * 2 points [-]

No, the details are very different from range voting. You still only allow each voter to pick one candidate, and the ultrafilter provides a (highly nonconstructive) way of distilling all of these votes into a winner of the election. You can get a feel for how this works by reading about what Terence Tao calls an adaptive oracle. Roughly speaking, you can arbitrarily choose winners subject to consistency with your previous choices and with the conditions in Arrow's theorem, and the freedom of having infinitely many voters ensures that this process won't lead to a contradiction.

In a certain sense, "ultrafilter voting" is still controlled by a sort of "virtual dictator," so even conceptually this isn't a strong blow to Arrow's theorem, and in practice "ultrafilter voting" is useless because you can't write down (non-principal) ultrafilters explicitly.

Comment author: SilasBarta comment score below threshold [+] (8 children)

Comment author: Qiaochu_Yuan 30 December 2012 03:15:28AM * 2 points [-]

Okay, then I don't understand what you're saying. What do you mean by "permitting (finite) voters to express preferences using the full spectrum of real numbers" when, as I've said, in "ultrafilter voting" each voter is still only allowed to pick one candidate instead of expressing a degree of preference between candidates?

Comment author: SilasBarta comment score below threshold [+] (6 children)

Comment author: Qiaochu_Yuan 31 December 2012 06:40:22AM 3 points [-]

I'm... not?

Comment author: SilasBarta 03 January 2013 10:01:55PM 0 points [-]

The voting on this thread is bizarre, but I think I know where some of my confusion comes from. I misread this comment:

No, the details are very different from range voting. You still only allow each voter to pick one candidate, and the ultrafilter provides a (highly nonconstructive) way of distilling all of these votes into a winner of the election.

I thought the bit about "you still only allow ... one candidate" was about range voting, and the ultrafilter clause was referring to theorem.

Nevertheless, while you make a lot of points that are interesting in isolation, I don't see how any of it is responsive to the question I asked, which was whether the range voting (and other systems like it) avoids the Arrow Theorem problems by allowing infinite (effective, virtual) voters.

To anyone still reading: I understand that I've made some logical errors in this thread, but why the severe downvoting for some honest questions? Is there a nicer way to tell someone it looks like they're changing the topic?

Comment author: Qiaochu_Yuan 03 January 2013 10:10:01PM 0 points [-]

I don't see how any of it is responsive to the question I asked, which was whether the range voting (and other systems like it) avoids the Arrow Theorem problems by allowing infinite (effective, virtual) voters.

The answer is still no. Again, the details are very different. If you studied them, it would be clear to you that the details are very different. I don't know what else there is to say. If you asked me "is an apple tasty because it behaves like an orange?" my answer would be "no, apples behave very differently from oranges" and I don't understand what would constitute a better answer than that.

Comment author: SilasBarta 03 January 2013 10:41:35PM * -1 points [-]

Actually, what happened here would be more like this exchange:

You: Red produce is exempt from the produce tariffs.
Me: Oh! Is that why they don't make you pay a tax on red carrots?
You: The details are different there. Apples have long been used in making cider, and cider has to be taxed, but it can't be double-taxed, and you can read about why [here]. Apples have a significantly different shape from carrots, and the necessity of certain infrastructure has led to shape influencing taxation.
Me: Well, I don't know about all those issues, but on the matter of whether red carrots being red gets them out of the produce tariffs, is my suggestion correct?
You: I don't even know what you're saying now. What do you mean by "red carrots being red" when, as I've said, apples can be used for cider? Look, just study tax law, I can't conceive of how I can provide any other kind of reply.

That's a simple substitution of what happened here:

You: Infinite-voter systems avoid the consequences of Arrow's Theorem.
Me: Oh, is that why range voting avoids it?
You: The details are different there. Ultrafilter voting provides a way to select a winner, similar to how an adaptive oracle works. It allows you to choose winners in a way that satisfies the Arrow constraints.
Me: Well, I don't know about all those issues, but on the matter of whether range voting being effectively infinite-voter (via infinite precision), and infinite-voter systems avoiding the consequences of Arrow's Theorem, am I right?
You: What do you mean "being effectively infinite voter ... infinite precision", when ultrafilter voting has you just vote for one candidate? Look, just study the topic, I can't conceive of how I could reply differently.

Comment author: Qiaochu_Yuan 03 January 2013 10:45:24PM * 0 points [-]

No, it really isn't. A closer analogy is that produce that is either red or orange is exempt from the produce tariffs, ultrafilter voting is red, and range voting is orange. If you aren't willing to study the details then I am not going to respond any further. I've paid 10 karma to respond now entirely because I think ultrafilters are great and people should learn about them, but they have nothing to do with range voting.

Comment author: SilasBarta 03 January 2013 10:57:29PM -2 points [-]

So were you wrong to say that all "infinite voter systems avoid the consequences of Arrow's Theorem" ("both red and orange are exempt"), or were you wrong to reject my point about range voting's infinite effective voters being proof that it avoids the consequences of Arrow's Theorem?

On the one hand, you want to say that the broad principle is true ("apples and oranges are exempt"/"all infinite voter systems are exempt"), but on the other hand, you don't want to agree that the broad principle has the implications I suggested ("Navals are oranges and thus exempt"/"Range voting has infinite voters as is thus exempt").

And a resolution of that inconsistency really does not require a thorough review of the cited sources.

Qiaochu_Yuan comments on Godel's Completeness and Incompleteness Theorems - Less Wrong

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