The ineffable number

Originally posted by Acolyte
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.

OK, the ineffable number is the smallest integer that cannot be described in less than twenty words.

Originally posted by Acolyte
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.

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...

Originally posted by Lord Rahl
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...

Consider the set of numbers {1-1/n : n = 1,2,3,...}. Which of these is the largest? 😛

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. 😞

Originally posted by Acolyte
Consider the set of numbers {1-1/n : n = 1,2,3,...}. Which of these is the largest? 😛

Ok, I'll give you that one 😀