Date: Jan 3, 2013 9:32 PM Author: ross.finlayson@gmail.com Subject: Re: Uncountable Diagonal Problem On Jan 3, 9:07 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>

wrote:

> On Jan 2, 12:48 am, Virgil <vir...@ligriv.com> wrote:

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> > In article

> > <de9ee3af-0823-4a99-8216-7b6033235...@po6g2000pbb.googlegroups.com>,

> > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

>

> > > On Jan 1, 11:22 pm, Virgil <vir...@ligriv.com> wrote:

> > > > In article

> > > > <ef09c567-1637-46b8-932a-bcb856e41...@r10g2000pbd.googlegroups.com>,

> > > > "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

>

> > > > > On Jan 1, 8:59 pm, Virgil <vir...@ligriv.com> wrote:

> > > > > > In article

> > > > > > <5e016173-aa1b-4834-9d70-0c6b08f19...@jl13g2000pbb.googlegroups.

> > > > > > com>, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:

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> > > > > > > On Jan 1, 7:29 pm, Virgil <vir...@ligriv.com> wrote:

> > > > > > > > In article But in that proof Cantor does not require a well

> > > > > > > > ordering of the reals, only an arbitrary sequence of reals

> > > > > > > > which he shown cannot to be all of them, thus no such

> > > > > > > > "counting" or sequence of some reals can be a count or

> > > > > > > > sequnce of all of them. --

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> > > > > > > Basically

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> > > > > > Nonsense deleted! --

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> > > > > Nonsense deleted, yours?

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> > > > Nope! --

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> > > Great: from demurral to denial.

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> 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.

> To whit: in ZFC there are and there aren't uncountably many

> intervals.

> Then, with regards to Cantor's first for the well-ordering of the

> reals instead of mapping to a countable ordinal, there are only

> countably many nestings in as to where then, the gap is plugged (or

> there'd be uncountably many nestings). Then, due properties of a well-

> ordering and of sets defined by their elements and not at all by their

> order in ZFC, the plug can be thrown to the end of the ordering, the

> resulting ordering is a well-ordering. Ah, then the nesting would

> still only be countable, until the plug was eventually reached, but,

> then that gets into why the plug couldn't be arrived at at a countable

> ordinal. Where it could be, then the countable intersection would be

> empty, but, that doesn't uphold Cantor's first proper, only as to the

> finite, not the countable. So, the plug is always at an uncountable

> ordinal, in a well-ordering of the reals. (Because otherwise it would

> plug the gap in the countable and Cantor's first wouldn't hold.)

>

> Then, that's to strike this:

> "So, there couldn't be uncountably many nestings of the interval, it

> must be countable as there would be rationals between each of those.

> Yet, then the gap is plugged in the countable: for any possible value

> that it could be. This is where, there aren't uncountably many limits

> that could be reached, that each could be tossed to the end of the

> well-ordering that the nestings would be uncountable. Then there are

> only countably many limit points as converging nested intervals, but,

> that doesn't correspond that there would be uncountably many limit

> points in the reals. "

> Basically that the the gap _isn't_ plugged in the countable.

>

> Then, there are uncountably many nested intervals bounded by

> irrationals, and there aren't.

>

Point being there are uncountably many disjoint intervals defined by

the irrationals of [0,1]: each non-empty disjoint interval contains a

distinct rational. Thus, a function injects the irrationals into a

subset of the rationals.

Regards,

Ross Finlayson