(Untitled)

[further_on]

in conversing with certain strong acquaintances of mine earlier this evening (arguably the first of two thousand seven, although the starting point is quite arbitrary) we arrived - by way of a joke involving the word "farce" - at the idea of words that are what their definitions mean. the most perhaps obvious example of this is "word;" here are

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[screamphoenix]

i'd add "metaphor," because to my own unofficial idea of language, it's all metaphor. goedel's theorem doesn't really apply here, i think, because language isn't coherent like the mathematical systems he was busting up. goedel basically imported a version of the epimenides paradox ("this statement is a lie," put shortly)into principia mathematica. that system, like others of its kind, let people find "theorems" in it, and each theorem was a true thing. their goal, ultimately, was to make a system that contained every truth and only truths, and a lot of people thought the principia had a good chance of doing that. goedel's theorem (call it "n")was "theorem n is not in principia mathematica." the principia had to admit or disallow the theorem; systems like that can't do both and remain coherent, unlike langauge, which is to me an observer-defined system that doesn't necessarily answer to those observers (though sometimes it does), if you want to call that a system. if theorem n were excluded from the principia, there would be truths the system didn't contain, and it would be incomplete, even though it might, ideally, contain only truths. if the theorem were included, the theorem ("n is not in the p m") would be a false theorem though completely proper according to the p m. the system then would contain falsehoods as well as truths, although in that case it might contain every truth. so goedel proved that in systems like the p m, the catalog of theorems they can contain is either circumscribed but potentially entirely true, or it's got, possibly, all the truths but with falsehoods in there too.

it's not that simple, of course, and meta-theorems that account for the paradoxical theorem n could be put into to resolve, sort of, the problem, making a specific exception of sorts for n and preserving the p m's coherence. the problem with that is that goedel showed, i think in the same paper, that those meta-theorems, and the meta-theorems of those meta-theorems, are all vulnerable to the epimenides move he made before. there's no way around it for the p m or any system like it. a totally awesome and really entertaining book on this (and music, and a i, and escher, and zen, and others) is douglas hofstader's goedel, escher, bach. it's arranged in alternating chapters and dialoguesm and the dialogues are really ingenious. some are arranged like fugues, one is one big palindrome, etc etc. SORRY IF THIS HORNS IN ON YOUR RESEARCH BUT I FIGURED YOU WOULD LIKE IT.