Date: Jan 3, 2013 12:07 PM Author: ross.finlayson@gmail.com Subject: Re: Uncountable Diagonal Problem On Jan 2, 12:48 am, Virgil <vir...@ligriv.com> wrote:

> In article

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

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

>

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> > 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.

>

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.

Regards,

Ross Finlayson