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Topic: Matheology § 285
Replies: 84   Last Post: Jun 15, 2013 6:05 PM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: Matheology � 285
Posted: Jun 11, 2013 11:32 PM

Zeit Geist <tucsondrew@me.com> wrote:

> On Tuesday, June 11, 2013 4:09:56 PM UTC-7, Ralf Bader wrote:
> > mueckenh@rz.fh-augsburg.de wrote:
> >
> >
> >

> > > On Tuesday, 11 June 2013 20:43:30 UTC+2, Zeit Geist wrote:
> >
> > >
> >
> > > > I think your algorithm fails at the limit stage.
> >
> > >
> >
> > > That's the same with enumerating the rationals and with the diagonal of
> >
> > > the list. No limits available in all these cases.
> >
> > > If we enumerate the rationals, we do it up to n (since there is no
> >
> > > infinite natural number).
> >
> >
> >
> > No. Maybe you try to do it so, because you are stupid. WE might do something
> >
> > like this: We note that a positive rational r can be written as
> >
> > r = p_1^n1 * p_2^n2 * ... * p_k^nk, i.e. a product of powers of primes p_i
> >
> > with integer exponents ni. And then we map r to p_1^f(n1) * p_2^(n2) * ...
> >
> > * p_k^f(nk), where f(n) = 2n if n>0 and f(n) = -2n-1 if n<0 or so. We take
> >
> > some care with the cases r=0, r=1, ni=0 and then we can show that we have
> >
> > got a bijection from the positive rationals to the naturals.
> >
> >
> >
> > There is nothing WE do "up to n". There is no limit involved. What we need
> >
> > are variables ranging over various sets of numbers. I don't know whether a
> >
> > set of numbers must "actually exist" in order to be allowed to use
> >
> > variables ranging over that set.
> >
> >
> >
> > And I don't care about any opinions philosophers might hold about this
> >
> > point. For example
> >
> >
> > is anything but convincing. There one reads:
> >
> > "2. The universe began to exist.
> >
> >
> >
> > 2.1 Argument based on the impossibility of an
> >
> > actual infinite.
> >
> >
> >
> > 2.11 An actual infinite cannot exist.
> >
> > 2.12 An infinite temporal regress of
> >
> > events is an actual infinite.
> >
> > 2.13 Therefore, an infinite temporal
> >
> > regress of events cannot exist. "
> >
> > Then some arguments for the alleged non-existence of an actual infinite are
> >
> > derived from Hilbert's hotel. But this is completely beyond the point,
> >
> > those arguments at best saying that the events in that infinite regress can
> >
> > not be shuffled around like the guests in that hotel. And indeed they
> >
> > can't, because they are in a fixed order and no events can b checked in or
> >
> > out of the past. (But even if the arguments given for the non-existence of
> >
> > actual infinities would apply, they still would not be convincing.)
> >

>
> You mention of Hilbert's Hotel gives me an idea of how to show that the
> rational numbers cannot be order isomorphic to the naturals. I don't have
> all the details, but here's a sketch of it.
>
> A bus containing all the rationals in [0,1] pulls up to Hilbert's Hotel.
> The Bellhop asks, "How did you all fit in that little bus?"
> Someone replies, "Well, we can pack ourselves in rather densely."
> ; )
>
> The rationals know they can fit in the Hotel. After they are only
> countably infinite. But they have a special request. They really
> like to be nearby their "close" friends. So they ask:
>
> "Can you guarantee that if I give you a real number, d, then you can
> give me a number number m, such that for any rational number, r,
> If a rational x is in ( r - d, r + d ),
> then x's room will be within m rooms of r's?
>
> The Bellhop looks at It and says
> "Wow, you guys really are dense."
>
> If we want an order isomorphism between the rationals and the
> naturals, the Topological property of "closeness" must be preserved.

The standard order for the naturals is a well-ordering.

The standard ordering for the rationals is a dense ordering.

No ordered set of more than one element can have both orderings, because
any such a dense ordering then produces an infinite decreasing sequence
which violates any well-ordering.
>
> I doubt, WM can prove his A then B then A then ... algorithm satisfies
> that requirement. I will working proving it can't.
>
>
> ZG

--

Date Subject Author
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Virgil
6/11/13 JT
6/11/13 Tucsondrew@me.com
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Virgil
6/11/13 Tucsondrew@me.com
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Tucsondrew@me.com
6/11/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 Virgil
6/11/13 Tucsondrew@me.com
6/11/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Tucsondrew@me.com
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Tucsondrew@me.com
6/14/13 mueckenh@rz.fh-augsburg.de
6/14/13 Tucsondrew@me.com
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Tucsondrew@me.com
6/15/13 Virgil
6/15/13 Tanu R.
6/15/13 mueckenh@rz.fh-augsburg.de
6/15/13 Virgil
6/15/13 Virgil
6/14/13 Virgil
6/13/13 Virgil
6/12/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/12/13 Virgil
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Virgil
6/11/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Scott Berg
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Virgil
6/12/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/11/13 LudovicoVan
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 LudovicoVan
6/11/13 mueckenh@rz.fh-augsburg.de
6/11/13 Tucsondrew@me.com
6/11/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Tucsondrew@me.com
6/12/13 mueckenh@rz.fh-augsburg.de
6/12/13 Virgil
6/12/13 Tucsondrew@me.com
6/12/13 Virgil
6/12/13 Virgil
6/13/13 mueckenh@rz.fh-augsburg.de
6/13/13 Virgil
6/12/13 Virgil
6/11/13 LudovicoVan
6/11/13 Virgil
6/11/13 Tanu R.
6/11/13 Tucsondrew@me.com
6/11/13 Tucsondrew@me.com
6/11/13 Tucsondrew@me.com
6/11/13 Virgil
6/11/13 Tanu R.
6/11/13 Virgil
6/11/13 Tanu R.