Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Uncountable Diagonal Problem
Posted:
Jan 3, 2013 5:17 PM
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In article
<9335ba48-224c-453a-81d5-383ca8cd22cb@px4g2000pbc.googlegroups.com>,
"Ross A. Finlayson" <ross.finlayson@gmail.com> 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.
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