Date: Jan 4, 2013 12:32 AM Author: Virgil Subject: Re: Uncountably Nested Intervals In article

<b0302dd5-6a04-4af0-9ae4-690cffd2633c@pd8g2000pbc.googlegroups.com>,

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

> On Jan 3, 7:02 pm, Virgil <vir...@ligriv.com> wrote:

> > In article

> > <b7e06477-b836-41b1-be03-c4d0fe3c2...@q16g2000pbt.googlegroups.com>,

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

> >

> >

> >

> >

> >

> >

> >

> >

> >

> > > 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:

> >

> > > > > 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.googlegroup

> > > > > > > > > s.

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

> >

> > > > > > > > > > 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. --

> >

> > > > > > > > > > Basically

> >

> > > > > > > > > Nonsense deleted! --

> >

> > > > > > > > Nonsense deleted, yours?

> >

> > > > > > > Nope! --

> >

> > > > > > 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.

> >

> > Yes there are, as I pointed out in a posting that Ross has carefully

> > snipped entirely.

> >

> > The set of intervals { [-x,x] : x is a positive irrational} is one

> > such set of uncountably many nested intervals bounded by irrationals.

> >

> > A simple, and obvious, example of what Ross claims does not exist.

> >

> >

> >

> > > 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.

> >

> > This too is false.

> > { [x,1-x] : x is an irrational between 0 and 1/2} being an explicit

> > counterexample. And as there are way more such intervals than rationals

> > in their union, no such injection from intervals as Ross claims to

> > rationals can exist.

> >

> > And Ross is totally wrong again!!!

> >

> > And Ross will, no doubt, snip all of this proof of his errors too, just

> > as he did the last one, if he repies at all.

> > --

>

> That example contains zero, a rational, no?

No! For z between 0 and 1/2, no interval from x to 1-x will contain 0.

>

> What, that is news? Once again your plain arguments against the man

> instead of for the argument show your lack of argumentative ability,

> and responsibility, and poor form.

Since my ARGUMENT was entirely a mathematical example refuting your own

mathematical claim, it is ad mathematics not ad hominem.

Though I did enjoy being able to show your mathematics to be totally

wrong!

> But, for me to note that, is it ad

> hominem, to note ad hominem?

It is certainly an ad hominem to claim it when it did not exist, as you

did.

> See, for that I would refrain: because

> it's less than perfectly ethical to argue ad hominem.

Particularly when you are in the wrong and trying to cover your ass.

> Also quit

> bullying me, I'm bigger than you. A suitable change of topic for the

> thread, to respect the time of readers, is more along the lines of

> "Uncountably Nested Intervals".

>

> Uncountably many nested intervals, each pairwise disjoint contains two

> rationals, or rather as nested their disjoint contains a rational.

"Uncountably many nested intervals, each pairwise disjoint"?

Nested intervals are not pairwise disjoint, at least in any real world.

>

> The rationals are dense in the reals. Deal with it.

What misleads Ross onto thinking I don't already?

Re the original issue:

There are countably infinite sequences of nested intervals with rational

endpoints. For example { (-1/n,1/n) : n in |N }, but obviously no

uncountable nested set of such sequences.

Ross then claimed that there could not be any uncountable set of nested

intervals with irrational endpoints, which is trivially false:

{ (-x, x) : x is a positive irrational} is just such an uncountable but

nested set of intervals with irrational endpoints as Ross had claimed

did not exist.

So Ross was wrong, and too chicken to own up.

--