In article <email@example.com>, "Ross A. Finlayson" <firstname.lastname@example.org> wrote:
> Seems clear enough: in ZFC, there are uncountably many irrationals, > each of which is an endpoint of a closed interval with zero. And, > they nest. Yet, there aren't uncountably many nested intervals, as > each would contain a rational.
While there is no SEQUENCE of uncountably may nested intervals, which the very definition of sequence prohibits, there are certainly SETS of uncountably many nested intervals.
EXAMPLE: For each real x in (0,1), [x, 2-x] is closed real interval and the set of such intervals is both nested and uncountable. But it is not a SEQUENCE of intervals.
So that what Ross thought was a paradox is just a kink in his thinker. --