Infinity : An interesting variant
apoorv
Consider a vase A with an infinite number of balls numbered from 1 onwards with a ball numbered w (omega) as well.We have another vase B which is empty.
Precisely at 00 hrs , we start transferring balls from vase A to vase B as follows:
Time Interval-------------Ball Number
[00,1/2)----------------------1
[1/2,1/2+1/4)-----------------2
[3/4,3/4+1/8)-----------------3
and so on.(We can presume that the transfer of the ball is complete just at the beginning of the corresponding interval.)
Precisely at 01hrs, we transfer the ball numbered w to the vase B.
For every initial interval of natural numbers, there is a precisely defined sub-time interval of [00,01) during which the numbers on the balls in vase B constitute that interval of natural numbers.The set in vase B during any interval [01,t) is {1,2,3. . .w}.
Thus, there does not appear to be any instant (leave alone an interval) when the contents of vase B are {1,2,3. . .}.
How does the transition from finite intervals to the infinite interval {1,2,3...w} take place without ever going through {1,2,3. . .}?
-Apoorv
A N Niel
<15199779.1142668883...@nitrogen.mathforum.org>,
apoorv <sudh...@hotmail.com> wrote:
> How does the transition from finite intervals to the infinite interval
> {1,2,3...w} take place without ever going through {1,2,3. . .}?
A discontinuous function need not pass through intermediate states when
it goes from one state to another.
What do you mean by "how"?
Maybe one could reply that discontinuous functions do not "really"
exist in nature...???
apoorv
> In article
> <15199779.1142668883...@nitrogen.math
> forum.org>,
> apoorv <sudh...@hotmail.com> wrote:
>
> > How does the transition from finite intervals to
> >the infinite interval
> > {1,2,3...w} take place without ever going through
> >{1,2,3. . .}?
>
> A discontinuous function need not pass through
> intermediate states when
> it goes from one state to another.
>
> What do you mean by "how"?
>
> Maybe one could reply that discontinuous functions do
> not "really"
> exist in nature...???
Perhaps you are saying that the function Ordinal(k)=I(k)={1,2,3. . .k} is discontinuous at w(infinity), with
limI(k)[k-->w-]={1,2,3. . .} and I(w)= lim I(k)[k-->w+]={1,2,3. . w}?
-Apoorv
Peter Webb
"apoorv" <sudh...@hotmail.com> wrote in message
news:15199779.1142668883...@nitrogen.mathforum.org...
w cannot be a ball. It is a limit ordinal, which means that it is a limit of
balls. It is not the last ball. This is akin to assuming that 1/0 = w, which
convenient though that would be, isn't the case.
Mike Kent
> Consider a vase A with an infinite number of balls numbered from 1 onwards
> with a ball numbered w (omega) as well.We have another vase B which is empty.
> Precisely at 00 hrs , we start transferring balls from vase A to vase B as follows:
> Time Interval-------------Ball Number
> [00,1/2)----------------------1
> [1/2,1/2+1/4)-----------------2
> [3/4,3/4+1/8)-----------------3
> and so on.
> Precisely at 01hrs, we transfer the ball numbered w to the vase B.
> For every initial interval of natural numbers, there is a precisely
> defined sub-time interval of [00,01) during which the numbers on the
> balls in vase B constitute that interval of natural numbers.
> Thus, there does not appear to be any instant (leave alone an interval)
> when the contents of vase B are {1,2,3. . .}
Stripping out all the stuff about balls time intervals
and transfers from one urn to another ...
Let t(n) = 1-2^n; then for each n, let
I(n) = { k | t(k) <= t(n) } = {1, ... , n}
for the symbol "w", let
I("w") = N = {1, 2, ...}
and for each real number s with 0 <= s <= 1, let
J(s) = [0,s] \intersect I(w)
Then J(1) = I("w"), but if 0 <= s < 1, then
there is an n in N for which J(s) = I(n).
There's nothing especially paradoxical about this.
The "paradox" in the problem as given is just that
a certain <i>physical</i> situation can't arise
from the process described, but then, the process
itself is physically impossible ...
Dave L. Renfro
> I am not sure whether this variant of the balls in vase
> problem been discussed before in this forum; so, in any
> case here it is.
Google, including books and scholar google searches,
the phrase "super tasks". Your variant is a well-known
example in philosophy.
Dave L. Renfro
Stephen J. Herschkorn
You need to be careful when you start talking about limits and
continuity. What is the codomain of your function? S(w) = {0, 1, 2,
..., w}, the successor of w? Then, under the order topology, it is
true that
lim I(k) [k -> w+] = a for any a in S(w), since no neighborhood of
w contains elements greater than w. If your codomain is some ordinal
a > S(w), then both lim I(k) [k -> w+] = w and lim I(k) [k -> w+] =
S(w). a is not T1 under the order topology.
As ANN points out, your original function f: [0,1] -> a is not
continuous either. lim f(x) [x -> 1-] = w; the limit from above can
be any value.
I prefer to avoid topology all together here. Suffice it to note that,
for any ordinal a > w,
sup{b in a: b < w} = w (This is also true when a = w.)
min{b in a: b > n for all n < w} = w
min{b in a: b > w} = S(w)
--
Stephen J. Herschkorn sjher...@netscape.net
Math Tutor on the Internet and in Central New Jersey and Manhattan
Stephen J. Herschkorn
Nonsense. Throw a ball in with the label w on it. Or, as the OP has
set it up, take a ball with the label 0, cross it out, and write w
on it instead.
Verdigris
This is a succinct demonstration of fighting fire with fire ie of infinity
overcoming itself through a process of unlimited (mental) acceleration.
I see no problem in the result that is not already inherent in the Cantorian
theory of infinity. It seems to me that the final interval is:
[t(u-1), t(u)], where t (u-1) = (r =1to r = w-1)Sum 2^-r. (nb: u-1 stands
for penultimate)
The inherent difficulty lies in Cantor's idea that the set of natural
numbers is a satisfactory yardstick. The final ball takes 2^-w hours to
transfer from vase A
to vase B, but the inclusion of ball zero would exceed your neat figure of 1
hour, as would the inclusion of an arbitrary number of extra balls in vase
A. (try recasting the model with extra balls).
The weakness of this model is that it assumes that there is only on
infinity, which continues only as far as the smallest rational fraction
1/2^w. The model subverts the conventional mathematical sum to 'infinity'
used in evaluating a declining geometric progression. This infinity is
covertly absolute infinity, not the one cowering below the symbol w. Your
model works for any finite or limited but infinite number of balls: the
issue of a zero time interval does not arise.
It would be interesting to replicate the mental experiment with two jars,
one containing w black balls (odd numbers) and the other w white balls (even
numbers). The black and white balls being transferred alternately to the
empty jar.
Kind regards
Tony Thomas
"apoorv" <sudh...@hotmail.com> wrote in message
news:15199779.1142668883...@nitrogen.mathforum.org...
apoorv
> If your codomain is some ordinal
> a > S(w), then both lim I(k) [k -> w+] = w and
> lim I(k) [k -> w+] =
> S(w).
-Apoorv
apoorv
> Google, including books and scholar google searches,
> the phrase "super tasks". Your variant is a
> well-known
> example in philosophy.
-Apoorv
cbr...@cbrownsystems.com
apoorv
> This is a succinct demonstration of fighting fire
> with fire ie of infinity
> overcoming itself through a process of unlimited
> (mental) acceleration.
> I see no problem in the result that is not already
> inherent in the Cantorian
> theory of infinity.
Now, at instant 1, we did nothing to change the set in the vase.So, the set [1,2,3...} was in the vase before instant 1.But, paradoxically, at no definable instant before 1, the set [1,2,3...} could have been in the vase.
-Apoorv
apoorv
> has a very nice overview.
Any specific reference to this one? The issue here is that the configuration {1,2,3. . .w} exists at 01 hrs. The configuration {1,2,3. . .} must exist before the configuration {1,2,3. .w} and hence must exist before 01 hrs. But at no definable instant before 01 hrs, it could exist.
-apoorv
cbr...@cbrownsystems.com
I must confess that I see no particular difference between claiming
that because the set {1,2,3,...} exists at 01Hrs, that it must
therefore have existed before 01Hrs; and that the given that the set
{1,2,3....} union {any element you might care to mention that is added
at 01Hrs, for example the set of object of the color yellow,} exists at
01Hrs, that one therefore can claim that the set {1,2,3...} must have
etc., etc.
Cheers - Chas
apoorv
> I must confess that I see no particular difference
> between claiming
> that because the set {1,2,3,...} exists at 01Hrs,
> that it must
> therefore have existed before 01Hrs; and that the
> given that the set
> {1,2,3....} union {any element you might care to
> mention that is added
> at 01Hrs, for example the set of object of the color
> yellow,} exists at
> 01Hrs, that one therefore can claim that the set
> {1,2,3...} must have
> etc., etc.
-Apoorv
Stephen J. Herschkorn
Sorry for the delay - I have been unable to get to sci.math for a few days.
My earlier post was in error. Of course an ordinal under the order
topology is Hausdorff, as is any space under the order topology.
Letting S be the successor function, S(k) approaches w as k
increases toward w, and S(a) approaches S(w) as a decreases
toward w.
apoorv
If we go to the ZF axioms, the axiom of infinity asserts the existence of a set that contains 0 (the null set) and all the successors of 0.Now this set is infinite if x U {x}=Sx !=x for all x. This is equivalent to x !e x for all x.
So this set is X={All x:x ! e x}.
If we consider the system ZF minus the powerset and regularity axiom, then a successor of X exists only if X !e X.But the set X is the Russell set and the question whether X e X or not cannot be decided.
-Apoorv