That Scamp Haruhi; Yuki and Axiomatic Set Theory
Sunday, May 24, 2009 6:58:05 PM
Of course when we talk of "Haruhi three years ago", we might mean "Haruhi now, as broadcast three years ago", or "the Haruhi being broadcast now, in events that happened when she was three years younger". Is it a coincidence that Kadokawa et al waited three years into the future before broadcasting events that happened three years in the past? Or were Kyoto Animation just busy making Lucky Star, K-On and Clannad?
Of course, when I say "of course" it is just a rhetorical device. In reality (by which I mean, in fiction), there is of course no "of course" with Haruhi.
But on to the main point, which is Yuki's cryptic comment about the antinomies of axiomatic set theory.
What Yuki actually meant to say was this (taken from The Mathematical Experience, p322, click for full size image if you can't read it)
In other words (my own, in fact), what Kyon is saying is that Mikuru's claim (in ep.3 or 5 of the original series (depending on whether you use Haruhi or Kyon ordering)) that people can be inserted into an alternate time frame without affecting subsequent time frames, is inconsistent with the "bamboo" incident where his meeting Haruhi three years ago has affected subsequent events, specifically it inspired Haruhi to come to North High School.
Yuki's response essentially uses the example of mathematics to note that even in the most logical of all endeavours, things don't necessarily add up. In mathematics, people always assumed certain fundamental principles were guaranteed true in themselves without having to refer to the real world. But then, things that had been seen as sacrosanct for over a thousand years, such as the principles of geometry, were one by one found to be dodgy (eg. due to the discovery of non-Euclidean geometry).
As a result, mathematicians cast around for new fundamental principles which might underpin mathematics before it sank into the mire of subjectivity and empiricism. One candidate for this was Set Theory. Sets are just groups of objects which can be defined by some common sense description, such as "girls with brightly coloured hair" or "numbers greater than zero". As such Set Theory seemed a useful generalisation of logic, which is also built up from combinations of simple expressions.
Unfortunately there was a Boogiepop Phantom in the woodpile. Bertrand Russell realised (as any linguist could have told him) that it was possible to make a pair of seemingly common sense descriptions which contradicted one another or even themselves, such as a set of sets which contains "all sets which contain themselves". When you then make the inverse of this, "the set of all sets which do no contain themselves", you get a paradox, since such a set neither contains, nor does not contain, itself.
So in summary, when Kyon says "this doesn't add up", and Yuki says "Axiomatic set theory ... contains antinomies", she is saying "shit happens, even in pure mathematics".
(Actually, I thought Koizumi made the point more elegantly when he pocketed the king from the chessboard so as to get out of check. As he noted, there was nothing illogical about his action, and yet it totally violated the rules of chess. In other words having one set of perfectly self-consistent rules doesn't preclude you from having another set of self-consistent rules which totally contradict the first set. Which ultimately is what the antinomies nonsense boils down to.)