Posers and Puzzles
20 Nov 04
Originally posted by AcolyteOK, the ineffable number is the smallest integer that cannot be described in less than twenty words.
That's like saying 'the number in [0,1] that squares to 10'. It's a bad definition because such a number doesn't exist; in other words, the set of numbers in [0,1] which cannot be described in less than twenty words has no least or gr ...[text shortened]... sets like that, though: take the open interval (0,1), for example.
Originally posted by AcolyteWhy? I might be missing something here, but I disagree with this. [0,1] has an upper bound. There is, surely, at least one number in this interval that satisfies the condition, ie cannot be descibed in 20 words, so there must be a largest?
That's like saying 'the number in [0,1] that squares to 10'. It's a bad definition because such a number doesn't exist; in other words, the set of numbers in [0,1] which cannot be described in less than twenty words has no least or greatest element.
Feel free to make me feel stupid now...
Originally posted by Lord RahlConsider the set of numbers {1-1/n : n = 1,2,3,...}. Which of these is the largest? 😛
Why? I might be missing something here, but I disagree with this. [0,1] has an upper bound. There is, surely, at least one number in this interval that satisfies the condition, ie cannot be descibed in 20 words, so there must be a largest?
Feel free to make me feel stupid now...
The set of natural numbers which cannot be described in under 20 words, however, leads to problems.
Suppose it's non-empty; then it has a least element x, because the natural numbers are well-ordered, with the well-ordering 1,2,3,... . But then x can de described in under 20 words, so the set must be empty. This is nonsense because there are only finitely many words, so the set of numbers which can be described in under 20 words is finite. 😞