Date: Jan 3, 2013 5:17 PM
Author: Virgil
Subject: Re: Uncountable Diagonal Problem
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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