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math, language, Richard's Paradox

"Clearly there are integers so huge they can’t be described in fewer than 22 syllables. Put them all in a big pile and consider the smallest one. It’s “the smallest integer that can’t be described in fewer than 22 syllables.”

That phrase has 21 syllables."

re: math, language, Richard's Paradox

@witchfynder_finder @noelle so what youʼre saying is, the set of integers which canʼt be described in fewer than N syllables is only well‑defined with the requirement that descriptions do not themselves make reference to that set

iʼd buy that

re: math, language, Richard's Paradox

@noelle ellie,,

math, language, Richard's Paradox

@noelle paradoxes are my favourite kinda mindgame because its moreso like looking at the entire argument and saying "there's something broken here but no one knows exactly what"

math, language, Richard's Paradox

@noelle@elekk.xyz the clear flaw in this is that it relies on the inherently vague application of "described" to create its paradox and I would posit that the initial premise upon which this logic is based, of there being integers that require a minimum of 22 syllables to be described, is in fact a fallacious claim

math, language, Richard's Paradox

@noelle@elekk.xyz that said, it's a cleverly constructed statement

re: math, language, Richard's Paradox

WARNING: METAMATHEMATICAL NERD SHIT AHEAD

well, there are an infinite number of integers, and a finite number of 21-syllable combinations, so clearly there has to be some integer that requires at least 22

and you can say a predicate P defines an integer if there exists exactly one integer that satisfies P

but it's a fairly classic result that for any theory of arithmetic T, there is no function f to translate statements about T to integers such that f(x) is true inside T if and only if T proves x

so as a consequence, you can't define 'definability' inside that theory, and the paradox falls apart, because it's really a paradox about how casual language lets you move between talking about statements inside a theory and statements about that theory

tl;dr: human language is weird and ill-suited for talking about math

re: math, language, Richard's Paradox

@hierarchon@inherently.digital @noelle@elekk.xyz ah, but even this highly technical point against the paradox misses one of the key features of the initial statement, which is the implied but not defined assumption of "described" meaning "uniquely described".

However, even if one takes that unstated condition as a given, the paradox does not maintain its paradoxical nature, as by proving that the phrase in question does not uniquely describe a single integer, it disqualifies itself.

👻™ Witchtholomew 🌩️@witchfynder_finder@cybre.spacemath, language, Richard's Paradox

@noelle i hATE THIS?????