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Topic: Potential Infinity (Re: Supertasks)
Replies: 547   Last Post: Jul 2, 2011 7:16 PM

 Messages: [ Previous | Next ]
 Apollo Posts: 34 Registered: 6/26/11
Re: Potential Infinity (Re: Supertasks) @Tony
Posted: Jun 27, 2011 4:11 AM

In article
Albrecht <albstorz@gmx.de> wrote:

> On 23 Jun., 22:43, Gus Gassmann <horand.gassm...@googlemail.com>
> wrote:

> > On Jun 23, 4:45 pm, Albrecht <albst...@gmx.de> wrote:
> >
> >
> >
> >
> >
> >
> >
> >
> >

> > > On 23 Jun., 20:28, Tony Orlow <t...@lightlink.com> wrote:
> >
> > > > On Jun 23, 11:42 am, Gus Gassmann <horand.gassm...@googlemail.com>
> > > > wrote:

> >
> > > > > This is your claim, right?
> >
> > > > > On Jun 19, 2:01 am, Albrecht <albst...@gmx.de> wrote:
> >
> > > > > > We may conjecture about the physical world. But the mental world is
> > > > > > crystal clear defined in that concern: there are no more numbers,
> > > > > > signs, words, books, thoughts, ... than denumerable many.

> >
> > > > > You elaborated on that:
> >
> > > > > > But you don't. So 1, 2, 3, ... is no infinity and e.g. aleph_0 is
> > > > > > not
> > > > > > a number after all numbers since there is no "after".
> > > > > > The idea that there is something after endlessness is alogical.

> >
> > > > > Uergil and I countered with the example: 0, 1/2, 2/3, ... where there
> > > > > very much *is* a number (i.e., 1) after all the other numbers. How is
> > > > > this example different? You offered the feeble

> >
> > > > > > You mix up convergence with divergence.
> >
> > > > > That this is a smokescreen is obvious:
> >
> > > > > On 23 Jun., 12:29, Gus Gassmann <horand.gassm...@googlemail.com>
> > > > > wrote:

> >
> > > > > > It is trivial to introduce on N a metric that makes the sequence 1,
> > > > > > 2,
> > > > > > 3, ... Cauchy. However, it does not converge in that metric to a
> > > > > > natural number. Introduce a new object omega and define it as the
> > > > > > limit of the sequence. End of story. Convergence/divergence simply
> > > > > > does not enter into whatever issues you have or pretend to have
> > > > > > with
> > > > > > set theory.

> >
> > > > > You did not respond to that. Why in your world is it OK to write 1/2,
> > > > > 2/3, 3/4, ... ,1 but not 1, 2, 3, ... , omega?

> >
> > > > > *OF COURSE* omega is not a natural number, but that is utterly beside
> > > > > the point. 1 is not a proper fraction, either.

> >
> > > > Hello. I'd like to make some comments.
> >
> > > > It appears at this point we are arguing about limits. As Albrecht
> > > > points out, 1-1/n, or (n-1)/n, has a limit *as n increases without
> > > > bound* of 1. As n increases *without bound*, 1/n decreases to 0, and
> > > > 1-1/n approaches 1. This is an identifiable real limit. As n increases
> > > > without bound, on the other hand, n does not have a limit, or "bound",
> > > > aleph_0 notwithstanding. There is a difference, and his distinction is
> > > > not without merit. There is no "bound" to N.

> >
> > > > On the other hand, while convergence depends on the difference between
> > > > two expressions approaching nothing, divergence depends on such a
> > > > difference increasing without bound, with some differences increasing
> > > > faster than others. Just as faster convergence results in a smaller
> > > > result, one may say the divergence correlates positively with the
> > > > result,making it larger. In other words, infinitary numbers defined as
> > > > divergent sequences may be compared, in Big-O fashion, and ordered in
> > > > "numerosity".

> >
> > > > Peace,
> >
> > > > Tony
> >
> > > You surely explain this facts better than I can since you are a native
> > > speaker, I think. The point is that there are unlimited many elements
> > > in convergent sequences. But we don't need all of them to show that
> > > they converge against a limit. It's enough to show that how much we
> > > examine of them ever, they approach to the limit better and better
> > > without going beyond. There is no totality of all of them required as
> > > some seems to suggest. In the case of divergent sequences there is no
> > > approach. The distance to the so-called limit aleph_0 or omega is
> > > always the same: infinite.

> >
> > This is the third time I have to correct this error.
> >
> > Let a distance function d on N (the set of natural numbers) be defined
> > by d(n,m) = |1/(n+1) - 1/(m+1)|. Then the sequence 1, 2, 3, ... is
> > Cauchy. (Do you need a proof of this?) Let omega be the limit of this
> > sequence and define d(n,omega) = 1/(n+1).
> >
> > Convergence and divergence has *NOTHING* to do with what we are
> > talking about.

>
> There is no error. If you change the subject there are other
> consequences. That's normal. But we don't talk about a metric on the
> naturals. We talk about the increasing sequence
>
> *
> **
> ***
> ****
> *****
> ...
>
> And this sequence isn't Cauchy, and wasn't Cauchy and will be never
> Cauchy.

At least not until an appropriate metric on the set of such sequences
is imposed, which a mnor modification of gus's above will accomplish.

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