Re: Length of elements in countable sets
From: Timothy Little (tim-via-n.i.net_at_little-possums.net)
Date: 02/25/05
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Date: 25 Feb 2005 23:36:57 GMT
John Bokma wrote:
> Moreoever, the poster wrote that it is possible to have a countable
> set consisting entirely of elements of finite length. To me this
> sounds as a contradiction :-D.
It's not a contradiction, but its negation is. I agree that it can be
counterintuitive.
Consider the sets X_k of all elements with length k. Obviously X_k is
nonempty for all k, and if we have a finite set of symbols than all
the X_k's are finite. Let Y = Union X_k. Now there are two cases:
1) If Y is finite, then Y has some number of elements n. But this
means that at most n of the sets X_k are nonempty, and the rest are
empty. That's a contradiction, so this case cannot be true.
2) If Y is infinite, then it is an infinite set where every element
has finite length.
- Tim
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