Re: INFINITY Revisited

On 26 Aug 2005 22:04:20 -0700, Don Whitehurst wrote:
> If at noon there are no balls remaining, does this have implications
> about the naturals having a one to one correspondence with the real
> decimalic numbers?

> At 2 minutes to noon the vase is empty.
> 0 => 0

> Consider A) at 1 minute to noon, balls 1 - 9 that are added to the vase
> have the following printed on their respective surfaces:

> 1 => 0.1
> 2 => 0.2
>: => :
> 9 => 0.9.
> This corresponds to all of the single digit decimals between 0 and 1.

> Consider also B) at 1/2 minute to noon, balls 10 - 99 that are added to
> the vase have the following printed on their respective surfaces:

> 10 => 0.01
> 11 => 0.11
[ ... ]

> At noon the set of natural numbers associated with the balls is
> infinite, and I believe all balls must have necessarily been removed.

Correct.

> By noon an infinite number of balls (with printing on the surface) has
> been added to the vase and removed. This seems to me to imply that by
> noon all balls with infinite decimalic representations such as 1/3 =
> 0.333..., Pi/10 = 0.31415..., and sqrt (2)/2 = 0.70710678..., as well
> as many that can't be denumerated in real life, must also have been
> placed in the vase and removed.

No. According to the scheme you have described, only the balls with
terminating decimal expressions have been added and removed.

> Consequently, if I am not mistaken, by the removal process all reals on
> the interval 0 to 1 were set in a one to one correspondence with the
> naturals or else the vase would not be empty. This occurred at noon
> when 1) the naturals stopped being a finite subset and became the
> infinite set of naturals and simultaneously when 2) the decimalic
> representations printed on the balls changed from terminating rationals
> into the repeating rationals and the irrationals. The terminating
> rationals at each succesive interval formed numerous series that were
> approahing an infinitely growing number of limit points.

> Does this really establish what Cantor's diagonal proof disproved??

No, it merely establishes that the set of terminating decimal expressions
is countable.


--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
.

Relevant Pages

  • Re: An uncountable countable set
    ... -1/n, where n is a natural number, there are balls in the vase. ... Let S be the set of naturals on balls removed before noon. ...
    (sci.math)
  • Re: An uncountable countable set
    ... makes the vase empty? ... So if you just keep on adding balls one at a time, ... You drew that from my suggestion of the number circle, ... and that's why any infinite set of naturals ...
    (sci.math)
  • Re: An uncountable countable set
    ... above) and given that the "moment the vase becomes empty" means the first time t>= -1 that Vis zero, then it follows that the "vase becomes empty" at t = 0. ... No balls are added or removed at noon, but the vase becomes empty at noon. ... The fact that you have no upper bound to the naturals. ...
    (sci.math)
  • Re: An uncountable countable set
    ... in the vase has a natural number on it, but at noon you say that there is a ball in the vase that does not have a natural number on it. ... To become empty means there is a change of state in the vase, from having balls to not having balls. ... Noon is not included in the experiment by the very fact that the ball numbers are limited to the standard naturals. ...
    (sci.math)
  • Re: An uncountable countable set
    ... one determines the /number/ of balls in the vase at time t for a ... labelled ball, for example the ball labelled 15, is in the vase at that ... You have no infinite iterations, so noon does not occur, at at every moment BEFORE noon there is a nonzero number of balls in the vase. ... If we had an upper bound on the set of naturals, then if n is a natural ...
    (sci.math)
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