Does the set of all sets that do not include themselves include itself?
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The essence of all these is the malformed self-referential definition. The surprise/interestingness/definition/inclusion is "defined" recursively in such a way as to be actually not well defined.
> the largest number that cannot be described in fewer than 100 characters
Almost certainly you mean the smallest number (= positive integer), no? (One doesn't automatically expect that the existence of some positive integer satisfying a property means that there is a largest such.)
"Theorem" version:
Every integer is interesting. (Otherwise, one number would be the smallest uninteresting integer. And that would be of notable interest.)
"Paradox" version:
What is the largest number that cannot be described in fewer than 100 characters?
Or even the classic paradox by Russell https://en.wikipedia.org/wiki/Russell%27s_paradox
Does the set of all sets that do not include themselves include itself?
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The essence of all these is the malformed self-referential definition. The surprise/interestingness/definition/inclusion is "defined" recursively in such a way as to be actually not well defined.