Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Uncountably Nested Intervals
Posted:
Jan 4, 2013 12:32 AM
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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.
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