Marble problem--put in 10 marbles, then remove 1
Steven
My friend and I have have different answers to the following problem. Who
is correct and why?
You have a countable number of marbles and a really big bag.
The marbles are labeled 1, 2, 3, ... and so on.
You put the first ten marbles into the aforementioned bag and then you take
marble number 1
out of the bag and discard it. Then you put marbles 11 through 20 into the
bag and then you
take marble number 2 out of the bag and discard it, and so on.
In the end how many marbles are in the bag?
Friend's Answer: This process does not have a well defined limit.
A sequence of sets only has a limit if the sequence is nested.
The sequence of sets A,B,C,... is nested means that either A > B > C>...
or A < B < C<...
In the first case the limit is their intersection and in the second case the
limit is their union.
A real line version of your marble problem would be like this:
What is the limit of the sequence of sets S_n = [n + 1,10n] ?
The answer is there is no limit.
S_n just travels down the real line towards positive infinity.
My answer: The bag is empty since if not then some marble, say numbered n,
is still in the bag. But this is not possible since in the nth iteration
marble number n was removed.
Thanks for your time.
Pubkeybreaker
within this newsgroup.
Arturo Magidin
Steven <sgott...@hotmail.com> wrote:
>
> My friend and I have have different answers to the following problem. Who
>is correct and why?
>
>You have a countable number of marbles and a really big bag.
>The marbles are labeled 1, 2, 3, ... and so on.
>You put the first ten marbles into the aforementioned bag and then you take
>marble number 1
>out of the bag and discard it. Then you put marbles 11 through 20 into the
>bag and then you
>take marble number 2 out of the bag and discard it, and so on.
>In the end how many marbles are in the bag?
>
>
>Friend's Answer: This process does not have a well defined limit.
Depends on your definition of limit.
It is clear that for every n, B(n) (the contents of the bag at step n)
is a subset of N. We can certainly ask about the intersection of B(n),
and reasonably call it a "limit" of the process.
>A sequence of sets only has a limit if the sequence is nested.
That's one definition of "limit" for sequences of sets. But it is not
the only one.
For example, a more general notion of "limit" for the sequence would
be:
Given a sequence of sets A_n, we define lim(A_n) to be the subset
of union(A_n) consisting of all elements x such that there exists
an N (which may depend on x) for which x lies in A_k for all k>N.
I.e.: given the sequence A(1), A(2), A(3), ..., A(n), ...
let B(0) = Union(A_i)
B(1) = Union(A_i); i>1
B(2) = Union(A_i); i>2
.
.
.
B(n) = Union(A_i); i>n.
.
.
.
Then B(i) is a nested sequence of sets, and we define the limit of the
A_i to be the intersection of the B(i).
>The sequence of sets A,B,C,... is nested means that either A > B > C>...
>or A < B < C<...
>In the first case the limit is their intersection and in the second case the
>limit is their union.
Under one possible interpretation of "limit of sets".
>A real line version of your marble problem would be like this:
> What is the limit of the sequence of sets S_n = [n + 1,10n] ?
Which could be answered using the process I gave above, in which case
the answer would be that the limit is empty.
>The answer is there is no limit.
Depending on your definition of limit. If you restrict "limit" to
nested sequences, yes. If you have more general definitions of limit,
no.
>S_n just travels down the real line towards positive infinity.
>My answer: The bag is empty since if not then some marble, say numbered n,
>is still in the bag. But this is not possible since in the nth iteration
>marble number n was removed.
You are essentially taking the view I outlined above for the meaning
of "limit". It is correct, as far as it goes, given that definition.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
mag...@math.berkeley.edu
patrick
> >In the end how many marbles are in the bag?
the process you described never ends, so any answer to the question is correct.
nos...@spamless.com
"Steven" <sgott...@hotmail.com> wrote in message
news:tz86f.10494$cg....@news02.roc.ny...
>
> My friend and I have have different answers to the following problem. Who
> is correct and why?
trivial, 10 - 1 = 9
So when you stop it is a multiple of 9.
Try adding a fewer number of marbles,- like 6.
Or taking out more -like 9.
What happens then?
Now go away.
David C. Ullrich
(Arturo Magidin) wrote:
>In article <tz86f.10494$cg....@news02.roc.ny>,
>Steven <sgott...@hotmail.com> wrote:
>>
>> My friend and I have have different answers to the following problem. Who
>>is correct and why?
>>
>>You have a countable number of marbles and a really big bag.
>>The marbles are labeled 1, 2, 3, ... and so on.
>>You put the first ten marbles into the aforementioned bag and then you take
>>marble number 1
>>out of the bag and discard it. Then you put marbles 11 through 20 into the
>>bag and then you
>>take marble number 2 out of the bag and discard it, and so on.
>>In the end how many marbles are in the bag?
>>
>>
>>Friend's Answer: This process does not have a well defined limit.
>
>Depends on your definition of limit.
I don't see what "limits" in whatever sense have to do with it.
Every marble that's added is also removed, so there are no marbles
remaining.
************************
David C. Ullrich
david petry
Steven wrote:
> My friend and I have have different answers to the following problem. Who
> is correct and why?
>
> You have a countable number of marbles and a really big bag.
> The marbles are labeled 1, 2, 3, ... and so on.
> You put the first ten marbles into the aforementioned bag and then you take
> marble number 1
> out of the bag and discard it. Then you put marbles 11 through 20 into the
> bag and then you
> take marble number 2 out of the bag and discard it, and so on.
> In the end how many marbles are in the bag?
"In the end" ?
The world of the completed infinite is a fictional world.
There is no truly "correct" answer. Most mathematicians will
tell you that "in the end" the bag is empty, and trying to
argue with them is a waste of time.
Arturo Magidin
David C. Ullrich <ull...@math.okstate.edu> wrote:
>On Fri, 21 Oct 2005 16:32:03 +0000 (UTC), mag...@math.berkeley.edu
>(Arturo Magidin) wrote:
>
>>In article <tz86f.10494$cg....@news02.roc.ny>,
>>Steven <sgott...@hotmail.com> wrote:
>>>
>>> My friend and I have have different answers to the following problem. Who
>>>is correct and why?
>>>
>>>You have a countable number of marbles and a really big bag.
>>>The marbles are labeled 1, 2, 3, ... and so on.
>>>You put the first ten marbles into the aforementioned bag and then you take
>>>marble number 1
>>>out of the bag and discard it. Then you put marbles 11 through 20 into the
>>>bag and then you
>>>take marble number 2 out of the bag and discard it, and so on.
>>>In the end how many marbles are in the bag?
>>>
>>>
>>>Friend's Answer: This process does not have a well defined limit.
>>
>>Depends on your definition of limit.
>
>I don't see what "limits" in whatever sense have to do with it.
I agree... up to a point.
>Every marble that's added is also removed, so there are no marbles
>remaining.
Yes: if you go on long enough, you will reach a point where any
specific marble is taken out.
But if you want to talk about what happens "in the end", "and so on", with
an infinite process, surely some sort of limiting process is needed to
discuss this "end state".
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
If every ball that has been put into the bag has also been removed by
"the end", then after the end what balls can remain?
This is not something that can take place in any physical reality, as in
any physical reality there cannot be infinitely many balls to begin with.
So it can only take place in a mathematical version of reality, as a
metaphysical gedankenexperiment. And in such mathematical worlds
mathematics rules.
david petry
In your metaphysical mathematics world, your are right and
consistent and wonderful. But others are free to create other
metaphysical mathematical worlds.
You may have the last word ...
Dave Seaman
That only makes things more difficult. To solve the problem directly
(without limits), you only need to answer one question: which balls are
left "in the end"?
If you introduce limits, you have to answer at least 3 questions:
(1) What sort of limit should we consider?
(2) What sort of value do we get for this limit?
(3) Does this limit agree with the answer that we would have gotten
without applying limits? That is, is the function in some sense
continuous "at the end"?
It seems to me that in order to answer question (3), you have to answer
the very same question as before, namely, what answer do you get without
applying limits at all? And if you fail to consider question (3), then
you simply have not answered the question that was asked.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
Arturo Magidin
What does "in the end" mean in a process that has no end? How does the
question you ask me to answer even begin to make sense without
addressing that point? To solve the problem "directly" as you suggest
requires you to interpret what "the end" means for a process with NO
end. You may be hiding it, but you are still considering limits of
some kind at the very instant you start talking about "in the end".
That said, since for most people it is intuitively clear what "in the
end" means for the purpose of this mental exercise, then for most
people it is indeed completely unnecessary (and only more confusing)
to introduce limits. Simply pointing out that ball n will be removed
in step n is enough to convince most people that there will be no ball
"in the end" (whatever that means).
>If you introduce limits, you have to answer at least 3 questions:
>
> (1) What sort of limit should we consider?
> (2) What sort of value do we get for this limit?
> (3) Does this limit agree with the answer that we would have gotten
> without applying limits? That is, is the function in some sense
> continuous "at the end"?
>
>It seems to me that in order to answer question (3), you have to answer
>the very same question as before, namely, what answer do you get without
>applying limits at all? And if you fail to consider question (3), then
>you simply have not answered the question that was asked.
I did not bring up limits to answer the question, in any case. The
original poster said his friend answered that there was "no answer"
because "the" (singular definitive article) limit of the sets in
question did not exist. I pointed out that even if you want to try to
interpret the question by invoking limits explicitly, his friend's
analysis was flawed because the kind of limits he was considering was
far too limited for the task at hand. If ->he<- really wants to
consider limits, he has to address your question (1), at which point I
pointed out a reasonable extension of ->his<- notion of limit that
would render ->his<- objection moot.
Virgil
"david petry" <david_lawr...@yahoo.com> wrote:
What constitutes "THE" metaphysical mathematics world, to the extent
that there is one at all, is a matter of general agreement, but is not
to be invalidated by particular disagreement.
Nathan the Living God
David C. Ullrich
(Arturo Magidin) wrote:
It means "after all the infinitely many steps have been carried
out".
Which of course is impossible in the real world. But saying
something about limits does not change the fact that in
the real world we can only do finitely many things before
we die - the problem _is_ an abstract mathematical thing
regardless.
The state of some system "after infinitely many steps"
need not always be well-defined. For example, you toggle
a light switch on and off infinitely many times; there
is no answer to the question of whether it's on or
off at the end. But that's not really because a certain
limit does not exist:
Q: Start with x = 1. At each step replace the value
of x with the current value of x divided by 2.
What is the value of x after infinitely many
steps?
If someone said the answer was 0 because
the limit of x was 0 I wouldn't feel inclined
to argue with that. But if someone said that
for _this_ problem "the value of x after infinitely
many steps" was undefined I wouldn't be inclined
to argue with _that_ either! Because the limit
of x_n is _not_ the same thing as "the value
of x after saying n = n+1, x = x_n infinitely
many times"...
In the current problem one could say that a certain
sequence of sets has an empty limit. But that
doesn't seem to me to be exactly what the problem
is asking, and the answer to exactly what the
problem is asking _is_ clear, without mentioning
limits: Every marble that is added is also removed
at some later stage, so after infinitely many
steps the jar is empty.
************************
David C. Ullrich
Dave Seaman
If each event has a time associated with it, then "in the end" can be the
least upper bound of the set of times. It's possible that this might
yield t=+oo, or perhaps t=w or higher (Google for "Transfinite Subway"
for an example it which the least upper bound is w_1).
Although the act of taking the least upper bound is a kind of limit
process, there is no limit involved in saying that if ball n is removed
at time t_n and is not subsequently replaced, then the ball remains
outside the bag for each time t > t_n.
David Hartley
<ITSnetNOTcom#virgil-586E21....@comcast.dca.giganews.com>,
Virgil <ITSnetNOTcom#vir...@COMCAST.com> writes
Here's a variation which is almost in physical reality. It doesn't
completely lose the infinite aspect, but pushes it back into a variation
of Zeno's paradox.
Suppose the hare and the tortoise are racing along a straight track of
length 1. The tortoise moves at a steady speed and completes the track
in 1 time unit. The hare starts out 10 times faster but decelerates
sharply and only just makes the finish. His movement is described by x =
1 - (1-t)^10, where x is the distance along the track, t the time.
Now label some points along the track.
Let Mn be the point with x = 1 - 1/2^n.
When the tortoise is at M1, the hare is at M10, when he is at M2, the
hare is at M20, and so on. After he passes Mn the M-points between them
correspond to the marbles in the jar after the n-th stage. Each time the
tortoise passes an M-point, the number of M-points between him and the
hare increases by 9. But they reach the finish together, and there are
then no M-points between them.
--
David Hartley
Rob Johnson
I have mentioned in previous threads on this topic that the topology
which is being assumed here is the topology of pointwise convergence
on the indicator functions of which marbles are in the bag. In the
context of this problem, the limit of the indicator functions is the
function which maps all marbles outside of the bag. This answers the
3 questions posed above by Dave Seaman.
As a topology on such indicator functions, pointwise convergence is
so natural that it is easy to see why some may think that topology is
unnecessary for this problem. However, it is this choice of topology
which admits this limit "in the end". Furthermore, saying a marble is
in the bag "in the end" if and only if it is in the bag at all times
past a certain point, is simply restating what convergence is under
the topology of pointwise convergence.
Rob Johnson <r...@trash.whim.org>
take out the trash before replying
Dave Seaman
No, it doesn't. It fails test (3), because we do not get agreement in
the case where balls are moved "at the end".
There is nothing in the solution I have presented that "assumes" a
topology of pointwise convergence, or any other convergence. In fact,
it's a consequence of my argument that the topology of pointwise
convergence fails for some versions of the problem.
The mere fact that we can apply test (3) at all is evidence that topology
is not needed to obtain the answer.
> As a topology on such indicator functions, pointwise convergence is
> so natural that it is easy to see why some may think that topology is
> unnecessary for this problem. However, it is this choice of topology
> which admits this limit "in the end". Furthermore, saying a marble is
> in the bag "in the end" if and only if it is in the bag at all times
> past a certain point, is simply restating what convergence is under
> the topology of pointwise convergence.
But again, you fail to account for balls in the bag at the instant when
they are added.
Larry Lard
Steven wrote:
> My friend and I have have different answers to the following problem. Who
> is correct and why?
>
> You have a countable number of marbles and a really big bag.
> The marbles are labeled 1, 2, 3, ... and so on.
> You put the first ten marbles into the aforementioned bag and then you take
> marble number 1
> out of the bag and discard it. Then you put marbles 11 through 20 into the
> bag and then you
> take marble number 2 out of the bag and discard it, and so on.
> In the end how many marbles are in the bag?
(There was an extemely long thread on this topic not very long ago at
all).
In passing I will note that the usual problem statement has the first
operation at some time, say 10 minutes before noon; the second
operation at 1 minute before noon; the third at 0.1 minute before noon;
and so on, with each operation taking place instantaneously. The
question is then posed as 'how many balls in the bag at 1 minute AFTER
noon'. This might look like a clever way to pin down what 'in the end'
means for an infinite process, but in practice just gets the physics
cranks involved as well as the maths cranks, by producing discussion
about the energy requirements etc etc.
--
Larry Lard
Replies to group please
Rob Johnson
I thought "at the end" (where I am using the topology of pointwise
convergence and you are using no topology, therefore, no limits), we
are both saying that there are no marbles in the bag.
>There is nothing in the solution I have presented that "assumes" a
>topology of pointwise convergence, or any other convergence. In fact,
>it's a consequence of my argument that the topology of pointwise
>convergence fails for some versions of the problem.
>
>The mere fact that we can apply test (3) at all is evidence that topology
>is not needed to obtain the answer.
Well, that is not totally true. What I assumed that you had meant by
comparing answers was that we both arrived at no marbles in the bag.
I claim that without using limits, there is no way to correctly deduce
an answer.
When you say that a marble is in or out of the bag "at the end" if and
only if it is in or out of the bag at all steps past some given step
(possibly a different step for each marble), then you are saying that
the indicator function of the marbles in the bag converges pointwise.
That is simply the definition of pointwise convergence. Therefore, you
are working under the topology of pointwise convergence.
>> As a topology on such indicator functions, pointwise convergence is
>> so natural that it is easy to see why some may think that topology is
>> unnecessary for this problem. However, it is this choice of topology
>> which admits this limit "in the end". Furthermore, saying a marble is
>> in the bag "in the end" if and only if it is in the bag at all times
>> past a certain point, is simply restating what convergence is under
>> the topology of pointwise convergence.
>
>But again, you fail to account for balls in the bag at the instant when
>they are added.
Here is the indicator function for the first several steps:
{}
{1,2,3,4,5,6,7,8,9,10}
{2,3,4,5,6,7,8,9,10}
{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
How does this fail to account for the balls in the bag at the instant
when they are added? This sequence of indicator functions converges
pointwise to {}.
Dave Seaman
> In article <djiqrg$h5m$1...@mailhub227.itcs.purdue.edu>,
> Dave Seaman <dse...@no.such.host> wrote:
>>> I have mentioned in previous threads on this topic that the topology
>>> which is being assumed here is the topology of pointwise convergence
>>> on the indicator functions of which marbles are in the bag. In the
>>> context of this problem, the limit of the indicator functions is the
>>> function which maps all marbles outside of the bag. This answers the
>>> 3 questions posed above by Dave Seaman.
>>No, it doesn't. It fails test (3), because we do not get agreement in
>>the case where balls are moved "at the end".
> I thought "at the end" (where I am using the topology of pointwise
> convergence and you are using no topology, therefore, no limits), we
> are both saying that there are no marbles in the bag.
Not in the case where marbles are added at that instant.
>>There is nothing in the solution I have presented that "assumes" a
>>topology of pointwise convergence, or any other convergence. In fact,
>>it's a consequence of my argument that the topology of pointwise
>>convergence fails for some versions of the problem.
>>The mere fact that we can apply test (3) at all is evidence that topology
>>is not needed to obtain the answer.
> Well, that is not totally true. What I assumed that you had meant by
> comparing answers was that we both arrived at no marbles in the bag.
> I claim that without using limits, there is no way to correctly deduce
> an answer.
You have not addressed the three questions that I presented when I
explained my objection to using limits.
(1) What sort of limit should we consider?
(2) What sort of value do we get for this limit?
(3) Does this limit agree with the answer that we would have gotten
without applying limits? That is, is the function in some sense
continuous "at the end"?
As I said, you cannot claim to have answered the question that was posed
unless you have answered question (3). But if you can answer question
(3), then questions (1) and (2) and the whole question of limits become
irrelevant. Even if you find a type of limit that gives an affirmative
answer to (3) in every case (which pointwise convergence does not), you
still can't claim that limits are in any sense necessary to finding the
answer. The answer comes first.
> When you say that a marble is in or out of the bag "at the end" if and
> only if it is in or out of the bag at all steps past some given step
> (possibly a different step for each marble), then you are saying that
> the indicator function of the marbles in the bag converges pointwise.
> That is simply the definition of pointwise convergence. Therefore, you
> are working under the topology of pointwise convergence.
The indicator function may or may not converge pointwise, depending on
how many times the marble is moved. Even if the indicator function
converges, it gives the wrong value in the case where a marble is moved
precisely "at the end" (at noon, perhaps, in one version of the problem).
>>> As a topology on such indicator functions, pointwise convergence is
>>> so natural that it is easy to see why some may think that topology is
>>> unnecessary for this problem. However, it is this choice of topology
>>> which admits this limit "in the end". Furthermore, saying a marble is
>>> in the bag "in the end" if and only if it is in the bag at all times
>>> past a certain point, is simply restating what convergence is under
>>> the topology of pointwise convergence.
Every question has an answer that is simple, obvious, natural, and wrong.
>>But again, you fail to account for balls in the bag at the instant when
>>they are added.
> Here is the indicator function for the first several steps:
> {}
> {1,2,3,4,5,6,7,8,9,10}
> {2,3,4,5,6,7,8,9,10}
> {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
> {3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
> How does this fail to account for the balls in the bag at the instant
> when they are added? This sequence of indicator functions converges
> pointwise to {}.
You fail to distinguish between the *value* of a function and its
*limit*. If the first 10 balls are added at time t=1, then the *value*
of the indicator function at t=1 indicates that balls 1-10 are in the
bag, but the *limit* as t->1 from the left indicates that the bag is
empty.
If you really intended to talk about *values* instead of *limits*, then you
are not using topology.
Star...@earthlink.net
By the nth iteration, n+1-n^2 will also be in the bag. So given we take
the limit to infinity. The infinitieth ball will not be in there, but
the infinity+1-10*infinity ones will be there.
That is, for iteration n, (n-1) through (10n) will be in the bag. This
yields 9n marbles. Just because n marbles will be removed and n
approaches infinity, and 10n approaches infinity too does NOT mean that
there will be no marbles.
You desperately need to take calculus classes. Using limit theorum,
then since the equation yields 9n marbles for n iterations, it matters
only that the equation yields 9n marbles for n iterations, and
therefore there will be an infinite number in the bag since as n tends
to infinity, so does 9n tend to infinity.
The amount left in the bag outweighs the amount taken out.
mensa...@aol.com
Star...@Earthlink.net wrote:
> How could there be no marbles left?
Because they were all taken out?
>
> By the nth iteration, n+1-n^2 will also be in the bag. So given we take
> the limit to infinity. The infinitieth ball will not be in there, but
> the infinity+1-10*infinity ones will be there.
Wait a minute...isn't infinity+1 equal to infinity?
So when you've taken out the infinitieth ball haven't you
also taken out the inifinitieth+1 ball? Likewise for 10*infinity?
>
> That is, for iteration n, (n-1) through (10n) will be in the bag. This
> yields 9n marbles. Just because n marbles will be removed and n
> approaches infinity, and 10n approaches infinity too does NOT mean that
> there will be no marbles.
No comprende.
>
> You desperately need to take calculus classes. Using limit theorum,
> then since the equation yields 9n marbles for n iterations, it matters
> only that the equation yields 9n marbles for n iterations, and
> therefore there will be an infinite number in the bag since as n tends
> to infinity, so does 9n tend to infinity.
If I list all the odd natural numbers and after each one extend
that number by multiplying by 2 an infinite number of times:
1 2 4 8 16 ...
3 6 12 24 48 ...
5 10 20 40 80 ...
7 14 28 56 112 ...
.
.
.
As I extend the number of rows to infinity, is there any
natural number not on the list?
> The amount left in the bag outweighs the amount taken out.
And I can conclude that there are infinitely more even numbers
than odd numbers?
Han de Bruijn
> The world of the completed infinite is a fictional world.
> There is no truly "correct" answer. Most mathematicians will
> tell you that "in the end" the bag is empty, and trying to
> argue with them is a waste of time.
I would like to add that this Marble problem is quite analogous to the
"Balls in a Vase" problem, which has been at the origin of the ongoing
"Infinity" thread (which has been monopolized meanwhile by Tony Orlow):
http://groups.google.nl/group/sci.math/msg/d2573fcb63cbf1f0?hl=en&
Both problems are interesting because they serve as a watershed between
different kind of mathematicians. Anyway, if a mathematician's answer is
that the bag is empty, then you can classify him as mainstream Cantorian
mathematician, for sure.
Wherefore by their fruits ye shall know them (Matthew 7:20).
Han de Bruijn
Rob Johnson
Dave Seaman <dse...@no.such.host> wrote:
>On Tue, 25 Oct 2005 00:51:59 GMT, Rob Johnson wrote:
>> In article <djiqrg$h5m$1...@mailhub227.itcs.purdue.edu>,
>> Dave Seaman <dse...@no.such.host> wrote:
>>>> I have mentioned in previous threads on this topic that the topology
>>>> which is being assumed here is the topology of pointwise convergence
>>>> on the indicator functions of which marbles are in the bag. In the
>>>> context of this problem, the limit of the indicator functions is the
>>>> function which maps all marbles outside of the bag. This answers the
>>>> 3 questions posed above by Dave Seaman.
>
>>>No, it doesn't. It fails test (3), because we do not get agreement in
>>>the case where balls are moved "at the end".
>
>> I thought "at the end" (where I am using the topology of pointwise
>> convergence and you are using no topology, therefore, no limits), we
>> are both saying that there are no marbles in the bag.
>
>Not in the case where marbles are added at that instant.
What instant? There is no last step, so how can we speak of marbles
added at the last instant? Even if we were to entertain the idea of
marbles added add the last instant, what marbles would those be?
>>>There is nothing in the solution I have presented that "assumes" a
>>>topology of pointwise convergence, or any other convergence. In fact,
>>>it's a consequence of my argument that the topology of pointwise
>>>convergence fails for some versions of the problem.
>
>>>The mere fact that we can apply test (3) at all is evidence that topology
>>>is not needed to obtain the answer.
>
>> Well, that is not totally true. What I assumed that you had meant by
>> comparing answers was that we both arrived at no marbles in the bag.
>> I claim that without using limits, there is no way to correctly deduce
>> an answer.
>
>You have not addressed the three questions that I presented when I
>explained my objection to using limits.
>
> (1) What sort of limit should we consider?
> (2) What sort of value do we get for this limit?
> (3) Does this limit agree with the answer that we would have gotten
> without applying limits? That is, is the function in some sense
> continuous "at the end"?
>
>As I said, you cannot claim to have answered the question that was posed
>unless you have answered question (3). But if you can answer question
>(3), then questions (1) and (2) and the whole question of limits become
>irrelevant. Even if you find a type of limit that gives an affirmative
>answer to (3) in every case (which pointwise convergence does not), you
>still can't claim that limits are in any sense necessary to finding the
>answer. The answer comes first.
The original problem tells what marbles are in the bag at any finite
step. It never states what marbles are in the bag "at the end" so I
don't see how we can compare the "value" of the function "at the end"
when none is given. If you claim that the "value at the end" is no
marbles in the bag because each marble is removed and not replaced,
then you are taking the limit as defined by pointwise convergence, not
the value as given by the problem. Otherwise, what is the "value at
the end" and why?
>> When you say that a marble is in or out of the bag "at the end" if and
>> only if it is in or out of the bag at all steps past some given step
>> (possibly a different step for each marble), then you are saying that
>> the indicator function of the marbles in the bag converges pointwise.
>> That is simply the definition of pointwise convergence. Therefore, you
>> are working under the topology of pointwise convergence.
>
>The indicator function may or may not converge pointwise, depending on
>how many times the marble is moved. Even if the indicator function
>converges, it gives the wrong value in the case where a marble is moved
>precisely "at the end" (at noon, perhaps, in one version of the problem).
Each marble is moved twice; once into the bag, and once out of the bag.
The indicator functions are well defined, and converge pointwise. Since
there is no "value" given "at the end", I don't see how to compare that
with the limit.
The domain of the indicator functions is the positive integers, so
there is no t->1 from the left. The first several values of the
indicator function are given in my last post, starting with {}.
In any case, I can't talk about a value at the end, since that was
never specified; the values were only specified at finite steps.
Each marble is removed from the bag at some point and never returned.
To say that that implies there are no marbles in the bag at the end,
is to say that the indicator functions converge pointwise. This can
also be stated as: the function mapping the steps to the indicator
functions is continuous on the one point compactification of the
positive integers (which has one limit point, infinity).
Star...@earthlink.net
How could they be? There's still infinity left.
> Wait a minute...isn't infinity+1 equal to infinity?
>So when you've taken out the infinitieth ball haven't you
>also taken out the inifinitieth+1 ball? Likewise for 10*infinity?
Infinity isn't actually a real number. So it doesn't matter.
> No comprende.
Learn calculus!
>If I list all the odd natural numbers and after each one extend
>that number by multiplying by 2 an infinite number of times:
>1 2 4 8 16 ...
>3 6 12 24 48 ...
>5 10 20 40 80 ...
>.7 14 28 56 112 ...
>.
>.
>.
>
>
>As I extend the number of rows to infinity, is there any
>natural number not on the list?
Nope.
>And I can conclude that there are infinitely more even numbers
>than odd numbers?
Nope. Seriously, read a calculus textbook.
Dave Seaman
> In article <djk80i$c43$1...@mailhub227.itcs.purdue.edu>,
> Dave Seaman <dse...@no.such.host> wrote:
>>On Tue, 25 Oct 2005 00:51:59 GMT, Rob Johnson wrote:
>>> In article <djiqrg$h5m$1...@mailhub227.itcs.purdue.edu>,
>>> Dave Seaman <dse...@no.such.host> wrote:
>>>>> I have mentioned in previous threads on this topic that the topology
>>>>> which is being assumed here is the topology of pointwise convergence
>>>>> on the indicator functions of which marbles are in the bag. In the
>>>>> context of this problem, the limit of the indicator functions is the
>>>>> function which maps all marbles outside of the bag. This answers the
>>>>> 3 questions posed above by Dave Seaman.
>>>>No, it doesn't. It fails test (3), because we do not get agreement in
>>>>the case where balls are moved "at the end".
>>> I thought "at the end" (where I am using the topology of pointwise
>>> convergence and you are using no topology, therefore, no limits), we
>>> are both saying that there are no marbles in the bag.
>>Not in the case where marbles are added at that instant.
> What instant? There is no last step, so how can we speak of marbles
> added at the last instant? Even if we were to entertain the idea of
> marbles added add the last instant, what marbles would those be?
There are several variations on the problem that we have been discussing.
In some versions, "the end" occurs precisely at noon. In at least one
version (the transfinite subway), "the end" comes when the subway reaches
station w_1. All we need to know is that the times when balls (marbles,
passengers, ...) are moved form a linearly ordered set and that "the end"
corresponds to the least upper bound of the transaction times. In some
variations of the problem, there may be transactions occurring precisely
at "the end". In the transfinite subway, there are many "ends" (many
limit ordinals) along the way, and passengers may board or disembark at
any of them.
>>>>There is nothing in the solution I have presented that "assumes" a
>>>>topology of pointwise convergence, or any other convergence. In fact,
>>>>it's a consequence of my argument that the topology of pointwise
>>>>convergence fails for some versions of the problem.
>>>>The mere fact that we can apply test (3) at all is evidence that topology
>>>>is not needed to obtain the answer.
>>> Well, that is not totally true. What I assumed that you had meant by
>>> comparing answers was that we both arrived at no marbles in the bag.
>>> I claim that without using limits, there is no way to correctly deduce
>>> an answer.
For which marble n can I not deduce the position of marble n at the end,
without using limits?
>>You have not addressed the three questions that I presented when I
>>explained my objection to using limits.
>> (1) What sort of limit should we consider?
>> (2) What sort of value do we get for this limit?
>> (3) Does this limit agree with the answer that we would have gotten
>> without applying limits? That is, is the function in some sense
>> continuous "at the end"?
>>As I said, you cannot claim to have answered the question that was posed
>>unless you have answered question (3). But if you can answer question
>>(3), then questions (1) and (2) and the whole question of limits become
>>irrelevant. Even if you find a type of limit that gives an affirmative
>>answer to (3) in every case (which pointwise convergence does not), you
>>still can't claim that limits are in any sense necessary to finding the
>>answer. The answer comes first.
> The original problem tells what marbles are in the bag at any finite
> step. It never states what marbles are in the bag "at the end" so I
> don't see how we can compare the "value" of the function "at the end"
> when none is given. If you claim that the "value at the end" is no
> marbles in the bag because each marble is removed and not replaced,
> then you are taking the limit as defined by pointwise convergence, not
> the value as given by the problem. Otherwise, what is the "value at
> the end" and why?
No, I am not taking the limit as defined by pointwise convergence.
Consider the following problem.
A marble is placed in an empty bag at noon. How many marbles
are in the bag at noon?
Your answer: by pointwise convergence, the indicator
function converges to the empty set as t->noon from the
left. Therefore, the bag is empty at noon.
My answer: there is one marble in the bag at noon, namely,
the one that was added at noon.
>>> When you say that a marble is in or out of the bag "at the end" if and
>>> only if it is in or out of the bag at all steps past some given step
>>> (possibly a different step for each marble), then you are saying that
>>> the indicator function of the marbles in the bag converges pointwise.
>>> That is simply the definition of pointwise convergence. Therefore, you
>>> are working under the topology of pointwise convergence.
The fact that my answer in that case happens to agree with the one given
by pointwise convergence does not mean I used pointwise convergence in
deriving my answer. See the problem just above for a counterexample.
My junior high math teacher was always pointing out that just because a
method happens to give the correct answer, it doesn't prove the method is
right.
>>The indicator function may or may not converge pointwise, depending on
>>how many times the marble is moved. Even if the indicator function
>>converges, it gives the wrong value in the case where a marble is moved
>>precisely "at the end" (at noon, perhaps, in one version of the problem).
> Each marble is moved twice; once into the bag, and once out of the bag.
In one version of the problem.
> The indicator functions are well defined, and converge pointwise. Since
> there is no "value" given "at the end", I don't see how to compare that
> with the limit.
There is a value given at the end. Each indicator function (in this
version of the problem) is defined for all t. If we know the position of
each marble at the end, then we also know the contents of the bag at the
end.
>>> {}
>>> {1,2,3,4,5,6,7,8,9,10}
>>> {2,3,4,5,6,7,8,9,10}
>>> {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
>>> {3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
You contradict yourself here, since you say later on that the domain is
the one point compactification of the positive integers.
> In any case, I can't talk about a value at the end, since that was
> never specified; the values were only specified at finite steps.
A marble is placed in the bag at 11:59 and is not subsequently moved.
Where is the marble at 12:00? Do you really claim that this question
cannot be answered?
What if the original problem is posed in such a way that the n-th step
takes place at 1/2^n minutes before noon, with the end coming precisely
at noon? It is perfectly obvious that the position of each marble is
specified at noon, since we are not told of any marble being added to the
bag after it is removed.
I don't consider the version of the problem with t0 = +oo to be any
different from the version with t0 = 0 (noon). In each case, the
transaction times form an ordered set and "the end" is the least upper
bound of the set.
> Each marble is removed from the bag at some point and never returned.
> To say that that implies there are no marbles in the bag at the end,
> is to say that the indicator functions converge pointwise.
You have cause and effect reversed. The indicator functions converge
pointwise precisely because we can deduce that each one is constant on
some appropriate interval.
> This can
> also be stated as: the function mapping the steps to the indicator
> functions is continuous on the one point compactification of the
> positive integers (which has one limit point, infinity).
The indicator function f_n is continuous at the end, on the assumption
that ball n is not moved at time t=t0 (the end). But the sum over all of
the indicator functions is wildly discontinuous at the end.
When you say an indicator function is continuous at t=t0 (the end), you
are saying precisely that for each n:
(1) lim_{t->t0-} f_n(t) exists,
(2) f_n(t0) exists, and
(3) f_n(t0) = lim_{t->t0-} f_n(t).
But in making statements (2) and (3) you are agreeing with me that
f_n(t0) has a meaning completely apart from the limit expression in (1).
I rest my case.
Jiri Lebl
> In the current problem one could say that a certain
> sequence of sets has an empty limit. But that
> doesn't seem to me to be exactly what the problem
> is asking, and the answer to exactly what the
> problem is asking _is_ clear, without mentioning
> limits: Every marble that is added is also removed
> at some later stage, so after infinitely many
> steps the jar is empty.
There is another way to show that "limits" may not always give you the
answer to the problem. This problem can be phrased in terms of
functions on N. Take f_k(n) be 1 if the nth ball is in the bag at the
kth step and 0 otherwise. Take the counting measure on N. Then f_k
are L^1 functions on N (or L^p for any p). Now the answer to the
problem is gotten by the pointwise limit of these functions as k tends
to infinity (that is, there is a notion of limit that gives the answer
which is the zero function), on the other hand, considering the space
under the L^p topology or the uniform topology, the sequence is not
Cauchy and the norm in fact goes off to infinity.
I want to reinforce David's point; that is, not every notion of limit
can be used for every problem and different notions of limits give
different answers. Especially if some limit is undefined, then it most
likely just says that you're not solving the problem in the correct
way. Further since the pointwise limit is a trivial one in this case,
one need not even use limits and most likely any attempt at using them
is likely to yield a confusing answer.
So just because you have no limit in SOME topology doesn't mean that
the problem has no solution. On the other hand, just because you get
some sort of limit existing doesn't mean it is the solution to your
problem. If one can solve a problem with the least amount of
formalism, then one is likely to get least confused.
Jiri
Jiri Lebl
> which is the zero function), on the other hand, considering the space
> under the L^p topology or the uniform topology, the sequence is not
> Cauchy and the norm in fact goes off to infinity.
I forgot to give some physical interpretation of why one would want to
use for example the L^1 norm topology. This is the topology that would
give you the limit in terms of the count of the marbles in the bag. Or
perhaps the weight of the bag, so it is not totally just a formal
nonsense limit notion, but in fact L^1 is really what all the cranks
are using in their intuition about this problem and why they think that
the bag must have infinitely balls in it "in the end".
The uniform norm (L^oo) on the other hand gives you a binary: 1 or 0
meaning nonempty or empty state of the bag and so also has physical
interpretation. And also arrives at the wrong conclusion.
Jiri
Keith A. Lewis
>How could there be no marbles left?
Please specify a schedule for the marble operations, and the time you're
interested in. Then it will be clear.
>By the nth iteration, n+1-n^2 will also be in the bag. So given we take
>the limit to infinity. The infinitieth ball will not be in there, but
>the infinity+1-10*infinity ones will be there.
IIRC the marbles are numbered with positive integers. There is no marble
with the number "infinity" painted on it. Infinity is not an integer.
>The amount left in the bag outweighs the amount taken out.
But none of the marbles with integers on them are left.
--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.
Jesse F. Hughes
More nonsense.
"Cantorian" doesn't seem to mean much of anything, but neither of the
analyses of this thought experiment have a damn thing to do with the
theorem that |X| < |P(X)|.
--
Jesse F. Hughes
"I'm better than you, and you know it."
-- James Harris
mensa...@aol.com
Star...@Earthlink.net wrote:
> > Because they were all taken out?
>
> How could they be?
Because the removal rule implies that "infinity" means "all".
> There's still infinity left.
Not when "all" are removed.
>
> > Wait a minute...isn't infinity+1 equal to infinity?
> >So when you've taken out the infinitieth ball haven't you
> >also taken out the inifinitieth+1 ball? Likewise for 10*infinity?
>
> Infinity isn't actually a real number.
Why then, do you think there is such a thing as infinity+1
or infinity*10?
> So it doesn't matter.
If the bag contains an infinite set of marbles and I remove
"all", then the bag is empty. If I remove an infinite subset,
say, the odd numbered marbles, then the bag still contains
an infinite number of marbles. So yes, it doesn't matter
that infinity is not a number, it only matters whether you
remove "all" or "some" of the marbles. The problem, as stated,
removes "all" the marbles, so at the end, the bag is empty.
>
> > No comprende.
>
> Learn calculus!
I did, but luckily I've forgotten it all.
>
> >If I list all the odd natural numbers and after each one extend
> >that number by multiplying by 2 an infinite number of times:
>
> >1 2 4 8 16 ...
> >3 6 12 24 48 ...
> >5 10 20 40 80 ...
> >.7 14 28 56 112 ...
> >.
> >.
> >.
> >
> >
> >As I extend the number of rows to infinity, is there any
> >natural number not on the list?
>
> Nope.
>
> >And I can conclude that there are infinitely more even numbers
> >than odd numbers?
>
> Nope.
So if infinity*infinity is not greater than infinity,
why do you think 10*infinity is?
> Seriously, read a calculus textbook.
Reading it is one thing, understanding it is another.
david petry
Jesse F. Hughes wrote:
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
>
> > david petry wrote:
> >
> >> The world of the completed infinite is a fictional world.
> >> There is no truly "correct" answer. Most mathematicians will
> >> tell you that "in the end" the bag is empty, and trying to
> >> argue with them is a waste of time.
> > Anyway, if a mathematician's answer is
> > that the bag is empty, then you can classify him as mainstream Cantorian
> > mathematician, for sure.
> More nonsense.
>
> "Cantorian" doesn't seem to mean much of anything, but neither of the
> analyses of this thought experiment have a damn thing to do with the
> theorem that |X| < |P(X)|.
Jesse, you're getting to be a nasty sort of guy in your old age.
The anti-Cantorian view is that it only makes sense to talk about
a potential infinite. The Cantorian view requires one to think
about the infinite as a completed kind of thing, and also implies
that we somehow have access to knowledge about how that completed
infinity must behave, even though it doesn't correspond to anything
we can observe.
>From the point of view that infinity only has a potential existence,
the only interesting thing to say about the marbles in a bag
problem is that the bag continually gets fuller and fuller without
end.
> "I'm better than you, and you know it."
> -- James Harris
Sounds like something JFH would say.
david petry
David C. Ullrich wrote:
> I don't see what "limits" in whatever sense have to do with it.
> Every marble that's added is also removed, so there are no marbles
> remaining.
When discussing these completed infinity kinds of things, it
is utterly absurd to try to use common sense, but ...
In this marbles problem, things change one step at a time. So,
if every marble has been removed, there must be some step in
which the last marble was removed. But, we know that just before
that particular step, ten additional marbles were added, so we
must conclude that there are at least nine marbles still remaining.
With a little more analysis, we would find that there are even
more marbles remaining.
Ullrich will now point out that common sense is some silly kind
of thing which only cranks would rely on.
david petry
Jiri Lebl wrote:
> but in fact L^1 is really what all the cranks
> are using in their intuition about this problem and why they think that
> the bag must have infinitely balls in it "in the end".
I would venture to say that 99.999% of the population
are "cranks", from your point of view.
Rob Johnson
I am working on the problem posed by the original poster of the thread.
The only transfinite involved is the phrase "at the end" when referring
to an unending process.
To end the process at noon, we can embed the index space (the positive
integers) into [0,1] (e.g. by t = 1-1/n) and, since we wish to consider
the value "at the end", map infinity in the one point compactification
of the index space to 1 (noon). This leaves a lot of points in [0,1]
with undefined values. We can fix that up by deciding that at a point,
x, in [0,1) not covered by the embedded integers, the value is the same
as at the greatest point which is covered and not greater than x.
However, this does not tell us the value of the indicator function at
infinity (or 1 or noon) any more than before.
>>>>>There is nothing in the solution I have presented that "assumes" a
>>>>>topology of pointwise convergence, or any other convergence. In fact,
>>>>>it's a consequence of my argument that the topology of pointwise
>>>>>convergence fails for some versions of the problem.
>
>>>>>The mere fact that we can apply test (3) at all is evidence that topology
>>>>>is not needed to obtain the answer.
>
>>>> Well, that is not totally true. What I assumed that you had meant by
>>>> comparing answers was that we both arrived at no marbles in the bag.
>>>> I claim that without using limits, there is no way to correctly deduce
>>>> an answer.
>
>For which marble n can I not deduce the position of marble n at the end,
>without using limits?
For no marble can we deduce its position at the end without some added
information. The state of each indicator function at all finite steps
(which is all the problem gives us) does not tell us the state of the
indicator function at infinity.
A couple of additional premises can define, in special circumstances,
which marbles are in the bag "at the end":
A: if a marble is taken out of the bag at some finite step and never
returned, then it is outside the bag "at the end"
B: if a marble is placed into the bag at some finite step and never
removed, then it is inside the bag "at the end"
These seem to be simply common sense when applied to physical models,
but they are necessary to make the math reflect the physical model.
These two additional premises imply that the bag is empty "at the end".
However, these two additional premises are simply a restatement of
what it means for the indicator functions of the marbles in the bag
to converge pointwise. These two premises imply that the indicator
functions converge pointwise.
Simply by assuming these premises, we are applying the topology of
pointwise convergence problem.
No, you are misinterpreting what I am saying. What I am saying is that
the premises I mentioned above, that seem to be assumed implicitly to
reflect a physical model, imply pointwise convergence. By adding a
marble at noon, you are contradicting those premises. The original
problem does not state that anything happens to the bag "at the end",
only at the finite steps before the end (or 1 or noon).
>>>> When you say that a marble is in or out of the bag "at the end" if and
>>>> only if it is in or out of the bag at all steps past some given step
>>>> (possibly a different step for each marble), then you are saying that
>>>> the indicator function of the marbles in the bag converges pointwise.
>>>> That is simply the definition of pointwise convergence. Therefore, you
>>>> are working under the topology of pointwise convergence.
>
>The fact that my answer in that case happens to agree with the one given
>by pointwise convergence does not mean I used pointwise convergence in
>deriving my answer. See the problem just above for a counterexample.
>
>My junior high math teacher was always pointing out that just because a
>method happens to give the correct answer, it doesn't prove the method is
>right.
Your junior high math teacher is completely correct in this regard.
>>>The indicator function may or may not converge pointwise, depending on
>>>how many times the marble is moved. Even if the indicator function
>>>converges, it gives the wrong value in the case where a marble is moved
>>>precisely "at the end" (at noon, perhaps, in one version of the problem).
>
>> Each marble is moved twice; once into the bag, and once out of the bag.
>
>In one version of the problem.
Well, at least in the version that was stated by the original poster.
>> The indicator functions are well defined, and converge pointwise. Since
>> there is no "value" given "at the end", I don't see how to compare that
>> with the limit.
>
>There is a value given at the end. Each indicator function (in this
>version of the problem) is defined for all t. If we know the position of
>each marble at the end, then we also know the contents of the bag at the
>end.
Each indicator function is only defined for t before noon. To assume
that this implies the value at noon requires some sort of convergence.
I added the indicator function at infinity so that I could speak about
the value at the end. The indicator functions given by the problem are
those indexed by the positive integers. The indicator function at
infinity is the one we are trying to deduce from those given in the
problem.
>> In any case, I can't talk about a value at the end, since that was
>> never specified; the values were only specified at finite steps.
>
>A marble is placed in the bag at 11:59 and is not subsequently moved.
>Where is the marble at 12:00? Do you really claim that this question
>cannot be answered?
No, but it takes premise B to answer it.
>What if the original problem is posed in such a way that the n-th step
>takes place at 1/2^n minutes before noon, with the end coming precisely
>at noon? It is perfectly obvious that the position of each marble is
>specified at noon, since we are not told of any marble being added to the
>bag after it is removed.
As long as one assumes premise A, then we know the position of each
marble at noon.
>I don't consider the version of the problem with t0 = +oo to be any
>different from the version with t0 = 0 (noon). In each case, the
>transaction times form an ordered set and "the end" is the least upper
>bound of the set.
I don't consider them different either. They are both the same, just
squished or stretched a bit geometrically.
>> Each marble is removed from the bag at some point and never returned.
>> To say that that implies there are no marbles in the bag at the end,
>> is to say that the indicator functions converge pointwise.
>
>You have cause and effect reversed. The indicator functions converge
>pointwise precisely because we can deduce that each one is constant on
>some appropriate interval.
I am not proposing any cause and effect. I am simply stating that
premises A and B are the same as assuming pointwise convergence. It's
an equivalence, not cause and effect.
>> This can
>> also be stated as: the function mapping the steps to the indicator
>> functions is continuous on the one point compactification of the
>> positive integers (which has one limit point, infinity).
>
>The indicator function f_n is continuous at the end, on the assumption
>that ball n is not moved at time t=t0 (the end). But the sum over all of
>the indicator functions is wildly discontinuous at the end.
I am confused as to what the sum over all the indicator functions has
to do with the problem.
>When you say an indicator function is continuous at t=t0 (the end), you
>are saying precisely that for each n:
>
> (1) lim_{t->t0-} f_n(t) exists,
> (2) f_n(t0) exists, and
> (3) f_n(t0) = lim_{t->t0-} f_n(t).
>
>But in making statements (2) and (3) you are agreeing with me that
>f_n(t0) has a meaning completely apart from the limit expression in (1).
I did not say that the function was continuous. The first part of what
I said was deleted in the quote above, but what I said was
|In any case, I can't talk about a value at the end, since that was
|never specified; the values were only specified at finite steps.
|Each marble is removed from the bag at some point and never returned.
|To say that that implies there are no marbles in the bag at the end,
|is to say that the indicator functions converge pointwise. This can
|also be stated as: the function mapping the steps to the indicator
|functions is continuous on the one point compactification of the
|positive integers (which has one limit point, infinity).
>I rest my case.
Star...@earthlink.net
number then obviously you're going to get an marble with an arbitrarily
high number on it, and arbitarily high might as well be infinite in the
equation.
How could you be so stupid as to assume there would be no marbles left?
If the n'th marble isn't in there, then the (n+1)th marble will be
there up to the (10*n)th marble. You can't argue that those marbles
aren't in there. If n was an arbitrarily high number that was still an
integer, this is always true. So there are always 9n marbles in the bag
no matter how high n you choose.
If you don't want to work with infinities, then don't introduce them.
Goddamn.
Han de Bruijn
> David C. Ullrich wrote:
>
>> [ ... ] Every marble that is added is also removed
>> at some later stage, so after infinitely many
>> steps the jar is empty.
[ ... Jiri's massive overkill argument deleted ... ]
What I really want to know is, Jiri: do you think that the jar is empty?
My bet is that you do.
Han de Bruijn
Han de Bruijn
> theorem that |X| < |P(X)|.
Whew! How impressive! I have a theorem that |Y| > |Q(Y)| - 1 .
Han de Bruijn
Han de Bruijn
> [ ... ] so at the end, the bag is empty.
Another mathematician classified as a Cantorian. It's easy ...
Han de Bruijn
Jesse F. Hughes
> Jesse F. Hughes wrote:
>> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
>>
>> > david petry wrote:
>> >
>> >> The world of the completed infinite is a fictional world.
>> >> There is no truly "correct" answer. Most mathematicians will
>> >> tell you that "in the end" the bag is empty, and trying to
>> >> argue with them is a waste of time.
>
>> > Anyway, if a mathematician's answer is
>> > that the bag is empty, then you can classify him as mainstream Cantorian
>> > mathematician, for sure.
>
>> More nonsense.
>>
>> "Cantorian" doesn't seem to mean much of anything, but neither of the
>> analyses of this thought experiment have a damn thing to do with the
>> theorem that |X| < |P(X)|.
>
> Jesse, you're getting to be a nasty sort of guy in your old age.
>
> The anti-Cantorian view is that it only makes sense to talk about
> a potential infinite.
Did Cantor introduce actual infinities in mathematics? Or did he
instead work to clarify the notion of infinite set, which already
appeared in the literature (however vaguely)?
I don't really know the answer to this, but I strongly suspect that
other mathematicians had used "actually" infinite sets long before
Cantor. Why would people accept informal talk of infinitesimals but
balk at infinite sets? It would be nice if someone knowledgeable
about historical mathematics from Leibniz to Cantor could comment.
If I am right, then it is silly to call this position
"anti-Cantorian". I would suggest "finitist", but that term is
already reserved for somewhat more coherent philosophies than this.
> The Cantorian view requires one to think about the infinite as a
> completed kind of thing, and also implies that we somehow have
> access to knowledge about how that completed infinity must behave,
> even though it doesn't correspond to anything we can observe.
Yeah, yeah, yeah. The Greeks (at least Aristotle and Plato) didn't
think mathematics was about the observable. Neither do the
overwhelming majority of modern mathematicians. What evidence do you
have that nineteenth century mathematicians believed that this
observability condition (whatever it means) was important prior to
Cantor's dastardly deeds?
In a sense, this historical question is irrelevant. One should
justify the claim that mathematical objects ought to correspond in
some very loose sense to the observable stuff or just drop this line
entirely. But besides claiming that mathematics *should* be about the
observable, you've claimed that it *was* about the observable, prior
to the Cantorian fall.
As an aside: Why the fuss over infinite sets? Why not attack
dimensionless points? The argument against them seems analogous to
the argument against infinite sets. Maybe we should talk about
"potentially dimensionless points" instead of "actually dimensionless
points", since every physical thing we've observed has *some*
extension.
> From the point of view that infinity only has a potential existence,
> the only interesting thing to say about the marbles in a bag
> problem is that the bag continually gets fuller and fuller without
> end.
>> "I'm better than you, and you know it."
>> -- James Harris
>
> Sounds like something JFH would say.
Well, maybe to certain folk, sure.
--
"The needs of the many outweigh the needs of the few [...] I must
make the same choice as those who came before me without regard to the
impact today, for the sake of the children of humanity, the children
of tomorrow." -- JSH channels Spock and generic politicians everywhere
Jesse F. Hughes
> David C. Ullrich wrote:
>
>> I don't see what "limits" in whatever sense have to do with it.
>> Every marble that's added is also removed, so there are no marbles
>> remaining.
>
> When discussing these completed infinity kinds of things, it
> is utterly absurd to try to use common sense, but ...
>
> In this marbles problem, things change one step at a time. So,
> if every marble has been removed, there must be some step in
> which the last marble was removed.
What does "so" mean here?
--
Jesse F. Hughes
"Would you please stop talking and start talking?"
-- Vincent Price as the Saint
David C. Ullrich
<david_lawr...@yahoo.com> wrote:
>
>David C. Ullrich wrote:
>
>> I don't see what "limits" in whatever sense have to do with it.
>> Every marble that's added is also removed, so there are no marbles
>> remaining.
>
>When discussing these completed infinity kinds of things, it
>is utterly absurd to try to use common sense, but ...
>
>In this marbles problem, things change one step at a time. So,
>if every marble has been removed, there must be some step in
>which the last marble was removed.
No, that simply does not follow.
>But, we know that just before
>that particular step, ten additional marbles were added, so we
>must conclude that there are at least nine marbles still remaining.
>With a little more analysis, we would find that there are even
>more marbles remaining.
We find that?
>Ullrich will now point out that common sense is some silly kind
>of thing which only cranks would rely on.
No. Instead I'll ask you _which_ marble is remaining at the end.
Let's give them names: say M1 is the first marble inserted,
M2 is the second one inserted, etc.
You say that we "find" that there are marbles remaining.
So give us an example - is M1 remaining, is M2 remaining,
etc?
************************
David C. Ullrich
Russell
> Are you RETARDED?
Hey, lighten up. Infinities are tricky, and you seem
not to have learned yet to proceed with caution. Shoot
from the hip and you might find it's your own foot that
gets hit.
If you're taking this limit to an arbitrarily high
> number then obviously you're going to get an marble with an arbitrarily
> high number on it, and arbitarily high might as well be infinite in the
> equation.
Just look at that and see how many weasel words you
used. ("Obviously"..., "might as well"...) Moreover
you're using the word "limit" without defining it, and
I can't see any way it could be any of the standard
limits defined in mathematics. Step back, take a
moment and try to make what you said precise enough to
qualify as mathematics. You may find this a valuable
exercise.
In any case, there's no such thing as "a marble with an
arbitrarily high number on it." Each marble has a specific,
*not* arbitrary, finite number on it. And for each such
marble, you can calculate the exact time that it is removed.
That time will be before noon.
>
> How could you be so stupid as to assume there would be no marbles left?
He didn't assume it. He deduced it.
> If the n'th marble isn't in there, then the (n+1)th marble will be
> there up to the (10*n)th marble.
*When* are they there? Yes, they are there for some time
after marble n is removed, but eventually every one of
them gets removed -- and that all happens before noon. So
it can't be any of *those* marbles that remain at noon.
You'll have to look elsewhere.
You can't argue that those marbles
> aren't in there. If n was an arbitrarily high number that was still an
> integer, this is always true. So there are always 9n marbles in the bag
> no matter how high n you choose.
Before noon, that is true. At exactly noon, it is not.
Yes, that is counterintuitive. Infinities *are*
counterintuitive.
>
> If you don't want to work with infinities, then don't introduce them.
> Goddamn.
The mathematicians here (as well as certain others of us)
don't have any problems working with infinities. But we
know better than to use infinities where they don't belong.
In particular, no marble with an infinite number on it is
ever put into the bag. I hope you can see that this is so;
that might be the first step toward what I think should be
a very nice "Aha" experience for you.
Dave Seaman
> In article <djlc74$ud4$1...@mailhub227.itcs.purdue.edu>,
> Dave Seaman <dse...@no.such.host> wrote:
>>There are several variations on the problem that we have been discussing.
>>In some versions, "the end" occurs precisely at noon. In at least one
>>version (the transfinite subway), "the end" comes when the subway reaches
>>station w_1. All we need to know is that the times when balls (marbles,
>>passengers, ...) are moved form a linearly ordered set and that "the end"
>>corresponds to the least upper bound of the transaction times. In some
>>variations of the problem, there may be transactions occurring precisely
>>at "the end". In the transfinite subway, there are many "ends" (many
>>limit ordinals) along the way, and passengers may board or disembark at
>>any of them.
> I am working on the problem posed by the original poster of the thread.
> The only transfinite involved is the phrase "at the end" when referring
> to an unending process.
This is not the first time we have had this discussion, you know. I
believe we have discussed this at least once in the context of the
transfinite subway problem.
> To end the process at noon, we can embed the index space (the positive
> integers) into [0,1] (e.g. by t = 1-1/n) and, since we wish to consider
> the value "at the end", map infinity in the one point compactification
> of the index space to 1 (noon). This leaves a lot of points in [0,1]
> with undefined values. We can fix that up by deciding that at a point,
> x, in [0,1) not covered by the embedded integers, the value is the same
> as at the greatest point which is covered and not greater than x.
> However, this does not tell us the value of the indicator function at
> infinity (or 1 or noon) any more than before.
While we are "fixing things up", why don't we just include "the end" in
the fixing process? What is so special about "the end"?
Let's simplify the problem by considering just a single ball. The ball
is added to an empty bag at 11:59. How many balls are in the bag at
noon? Do you claim the answer is unspecified, because noon is "the end"
and the problem doesn't specify what happens at "the end"?
What if the problem asks about 11:59:30 instead of noon? Does that mean
the answer is specified, or is it still unspecified because 11:59:30 has
now become "the end" and you have decreed that "the end" requires special
treatment?
Suppose there are two balls, one added at 11:58 and one at 11:59. How
many balls are in the bag at noon? How about 11:58:30? Do you claim
that the reasoning is somehow different for those two times, because one
is "the end" and the other is not?
>>>>>>The mere fact that we can apply test (3) at all is evidence that topology
>>>>>>is not needed to obtain the answer.
>>>>> Well, that is not totally true. What I assumed that you had meant by
>>>>> comparing answers was that we both arrived at no marbles in the bag.
>>>>> I claim that without using limits, there is no way to correctly deduce
>>>>> an answer.
>>For which marble n can I not deduce the position of marble n at the end,
>>without using limits?
> For no marble can we deduce its position at the end without some added
> information. The state of each indicator function at all finite steps
> (which is all the problem gives us) does not tell us the state of the
> indicator function at infinity.
> A couple of additional premises can define, in special circumstances,
> which marbles are in the bag "at the end":
> A: if a marble is taken out of the bag at some finite step and never
> returned, then it is outside the bag "at the end"
> B: if a marble is placed into the bag at some finite step and never
> removed, then it is inside the bag "at the end"
I note that neither of your premises mentions limits or topology in any
way.
> These seem to be simply common sense when applied to physical models,
> but they are necessary to make the math reflect the physical model.
> These two additional premises imply that the bag is empty "at the end".
Still no mention of limits, and your proof is now complete. You have
just confirmed my premise that pointwise convergence is unnecessary to
the solution.
> However, these two additional premises are simply a restatement of
> what it means for the indicator functions of the marbles in the bag
> to converge pointwise. These two premises imply that the indicator
> functions converge pointwise.
I agree that pointwise convergence is a corollary of your argument, but
you have not used that corollary in the proof that you just presented.
> Simply by assuming these premises, we are applying the topology of
> pointwise convergence problem.
Where? You did not get around to mentioning convergence at all until
*after* you had completed the proof. That's a strange way of "applying"
something.
By the way, your premises are specialized to "the end", but they can be
restated to apply to any point in time in an interval where a marble is
not being moved. I see no difference between "the end" and any other
time when the generalized premises apply.
[Lots of repetition snipped.]
>>> This can
>>> also be stated as: the function mapping the steps to the indicator
>>> functions is continuous on the one point compactification of the
>>> positive integers (which has one limit point, infinity).
>>The indicator function f_n is continuous at the end, on the assumption
>>that ball n is not moved at time t=t0 (the end). But the sum over all of
>>the indicator functions is wildly discontinuous at the end.
> I am confused as to what the sum over all the indicator functions has
> to do with the problem.
This was an aside, not directly related to anything you said, but it
certainly is relevant to the original problem. The problem asks how many
marbles are in the bag at the end, and the function I just described (f =
sum_n f_n) gives precisely the cardinality of the set of marbles in the
bag as a function of time. What the problem asks for is f(t0) = f(the
end), and not lim_{t->t0-} f(t), which is quite different.
>>When you say an indicator function is continuous at t=t0 (the end), you
>>are saying precisely that for each n:
>> (1) lim_{t->t0-} f_n(t) exists,
>> (2) f_n(t0) exists, and
>> (3) f_n(t0) = lim_{t->t0-} f_n(t).
>>But in making statements (2) and (3) you are agreeing with me that
>>f_n(t0) has a meaning completely apart from the limit expression in (1).
> I did not say that the function was continuous. The first part of what
> I said was deleted in the quote above, but what I said was
>|In any case, I can't talk about a value at the end, since that was
>|never specified; the values were only specified at finite steps.
>|Each marble is removed from the bag at some point and never returned.
>|To say that that implies there are no marbles in the bag at the end,
>|is to say that the indicator functions converge pointwise. This can
>|also be stated as: the function mapping the steps to the indicator
>|functions is continuous on the one point compactification of the
>|positive integers (which has one limit point, infinity).
Do you really mean to say that f(t0) = 0 iff (An) lim_{t->t0-} f_n(t)
converges? I assume you meant to specify that each of the indicator
functions converges to zero, but that's still not sufficient, because the
indicator functions could be discontinuous at t0.
Your premises A and B prove to my satisfaction that the bag is empty at
the end, but your own argument draws that conclusion without invoking
limits, except by way of an addendum after the proof is complete.
Han de Bruijn
> Did Cantor introduce actual infinities in mathematics? Or did he
> instead work to clarify the notion of infinite set, which already
> appeared in the literature (however vaguely)?
The answer is: yes. Cantor is the guilty one par excellance.
> I don't really know the answer to this, but I strongly suspect that
> other mathematicians had used "actually" infinite sets long before
> Cantor.
You mean Galileo Galilei, with his equi-numerosity of the squares and
the naturals? That surely is amateurish, when compared with Cantor's
contribution to mathematical "science".
> Why would people accept informal talk of infinitesimals but
> balk at infinite sets?
Informal perhaps. But the formal talk includes _limits_ and there is
nothing questionable with that.
> It would be nice if someone knowledgeable
> about historical mathematics from Leibniz to Cantor could comment.
>
> If I am right, then it is silly to call this position
> "anti-Cantorian". I would suggest "finitist", but that term is
> already reserved for somewhat more coherent philosophies than this.
Anti-Cantorian is a very good nomer. And we are continuously improving
our methods. We can distinguish the Cantorians from the rest now by a
simple test: the subject of this thread.
> As an aside: Why the fuss over infinite sets? Why not attack
> dimensionless points? The argument against them seems analogous to
> the argument against infinite sets. Maybe we should talk about
> "potentially dimensionless points" instead of "actually dimensionless
> points", since every physical thing we've observed has *some*
> extension.
Whew! Good! A point is the limit of a thing that has *some* extension.
Han de Bruijn
Jiri Lebl
> [ ... Jiri's massive overkill argument deleted ... ]
>
> What I really want to know is, Jiri: do you think that the jar is empty?
> My bet is that you do.
Yes of course it is empty. The problem is that the statement of the
problem makes it sound like something that could actually happen in
real life. That's what gets all of you all riled up. If the problem
(identical logical statement) would be stated in purely abstract terms
without references to marbles, bags, jars or ping pong balls, nobody
would get all that angry. It is a mathematical thought experiment, not
a real life situation. Trying to apply either real life intuition or
perhaps physical laws is bound to cause confusion. It is to
demonstrate that you have to be careful in logical arguments involving
large sets (in this case N).
So yes the jar (bag or whatever) is empty after the infinitely many
steps. The weight (L^1 norm) of the jar keeps increasing to infinity
and after all steps are performed it is 0. Not every function is
continuous.
Jiri
Torkel Franzen
> Anti-Cantorian is a very good nomer.
It signifies crackpottery.
Pubkeybreaker
(1) Your comments and behavior make you look arrogant.
(2) Your "math" consists of handwaving gibberish and shows great
ignorance.
(3) You use the combination to denigrate others.
I will refrain from any perjoratives regarding how the combination of
(1), (2), and (3)
makes you look to outsiders.
A countably infinite number of marbles was placed in the bag.
A countably infinite number was removed.
Since ALL countably infinite sets have the same size, there are no
marbles
left. Every labelled ball placed in the bag was removed at some time.
Wait. Let me guess. You think that the set
A = {0,1,2,3,4,5,6,7,8,9,10,11,12....}
is "bigger" than the set
B = {0, 10, 20, 30, 40, ...}
Because the former contains elements that are not in the latter.
Right?
Well, I have news. Both sets are the same size. For each element in A
there is
one and only one corresponding element in B. And vice-versa.
mensa...@aol.com
Pubkeybreaker wrote:
> "Are you RETARDED? If you're taking this limit.... <snip"
>
> (1) Your comments and behavior make you look arrogant.
> (2) Your "math" consists of handwaving gibberish and shows great
> ignorance.
> (3) You use the combination to denigrate others.
>
> I will refrain from any perjoratives regarding how the combination of
> (1), (2), and (3)
> makes you look to outsiders.
>
> A countably infinite number of marbles was placed in the bag.
> A countably infinite number was removed.
>
> Since ALL countably infinite sets have the same size, there are no
> marbles
> left.
But they are not necessarily identical, are they?
Can't you remove an infinite subset of a countable infinity
and still have a countable infinity left in the bag? Isn't it
critical to this problem that the definition of what gets
removed is the entire set of natural numbers and not some subset,
such as the evens?
Keith A. Lewis
>Are you RETARDED?
Nice discourse, troll.
*plonk*
Rob Johnson
Dave Seaman <dse...@no.such.host> wrote:
>On Wed, 26 Oct 2005 01:42:55 GMT, Rob Johnson wrote:
>> In article <djlc74$ud4$1...@mailhub227.itcs.purdue.edu>,
>> Dave Seaman <dse...@no.such.host> wrote:
>>>There are several variations on the problem that we have been discussing.
>>>In some versions, "the end" occurs precisely at noon. In at least one
>>>version (the transfinite subway), "the end" comes when the subway reaches
>>>station w_1. All we need to know is that the times when balls (marbles,
>>>passengers, ...) are moved form a linearly ordered set and that "the end"
>>>corresponds to the least upper bound of the transaction times. In some
>>>variations of the problem, there may be transactions occurring precisely
>>>at "the end". In the transfinite subway, there are many "ends" (many
>>>limit ordinals) along the way, and passengers may board or disembark at
>>>any of them.
>
>> I am working on the problem posed by the original poster of the thread.
>> The only transfinite involved is the phrase "at the end" when referring
>> to an unending process.
>
>This is not the first time we have had this discussion, you know. I
>believe we have discussed this at least once in the context of the
>transfinite subway problem.
Yes, we have, but I was working on the problem at hand.
>> To end the process at noon, we can embed the index space (the positive
>> integers) into [0,1] (e.g. by t = 1-1/n) and, since we wish to consider
>> the value "at the end", map infinity in the one point compactification
>> of the index space to 1 (noon). This leaves a lot of points in [0,1]
>> with undefined values. We can fix that up by deciding that at a point,
>> x, in [0,1) not covered by the embedded integers, the value is the same
>> as at the greatest point which is covered and not greater than x.
>> However, this does not tell us the value of the indicator function at
>> infinity (or 1 or noon) any more than before.
>
>While we are "fixing things up", why don't we just include "the end" in
>the fixing process? What is so special about "the end"?
We can't include "the end", because the state "at the end" can not be
deduced from any of the preceding states unless we add some premises.
The problem tells how to go from one state to the succeeding state.
The problem says nothing about states which have no predecessor. To
describe what happens at a state indexed by a limit ordinal, we must
have more premises.
Furthermore, I don't see any reason to increase the domain from the
positive integers (or possibly their one-point compactification) by
adding a lot of real intervals on which nothing occurs. It may only
confuse the issue.
>Let's simplify the problem by considering just a single ball. The ball
>is added to an empty bag at 11:59. How many balls are in the bag at
>noon? Do you claim the answer is unspecified, because noon is "the end"
>and the problem doesn't specify what happens at "the end"?
>
>What if the problem asks about 11:59:30 instead of noon? Does that mean
>the answer is specified, or is it still unspecified because 11:59:30 has
>now become "the end" and you have decreed that "the end" requires special
>treatment?
>
>Suppose there are two balls, one added at 11:58 and one at 11:59. How
>many balls are in the bag at noon? How about 11:58:30? Do you claim
>that the reasoning is somehow different for those two times, because one
>is "the end" and the other is not?
In these examples, the last state is indexed by a successor ordinal.
The problem tells what happens in these cases. This is very different
than talking about an unending sequence of ever changing states.
>>>>>>>The mere fact that we can apply test (3) at all is evidence that topology
>>>>>>>is not needed to obtain the answer.
>
>>>>>> Well, that is not totally true. What I assumed that you had meant by
>>>>>> comparing answers was that we both arrived at no marbles in the bag.
>>>>>> I claim that without using limits, there is no way to correctly deduce
>>>>>> an answer.
>
>>>For which marble n can I not deduce the position of marble n at the end,
>>>without using limits?
>
>> For no marble can we deduce its position at the end without some added
>> information. The state of each indicator function at all finite steps
>> (which is all the problem gives us) does not tell us the state of the
>> indicator function at infinity.
>
>> A couple of additional premises can define, in special circumstances,
>> which marbles are in the bag "at the end":
>
>> A: if a marble is taken out of the bag at some finite step and never
>> returned, then it is outside the bag "at the end"
>
>> B: if a marble is placed into the bag at some finite step and never
>> removed, then it is inside the bag "at the end"
>
>I note that neither of your premises mentions limits or topology in any
>way.
"at the end" is infinity, the limit of the one-point compactification
of the positive integers. These two premises describe the topology of
pointwise convergence.
>> These seem to be simply common sense when applied to physical models,
>> but they are necessary to make the math reflect the physical model.
>> These two additional premises imply that the bag is empty "at the end".
>
>Still no mention of limits, and your proof is now complete. You have
>just confirmed my premise that pointwise convergence is unnecessary to
>the solution.
>
>> However, these two additional premises are simply a restatement of
>> what it means for the indicator functions of the marbles in the bag
>> to converge pointwise. These two premises imply that the indicator
>> functions converge pointwise.
>
>I agree that pointwise convergence is a corollary of your argument, but
>you have not used that corollary in the proof that you just presented.
Pointwise convergence is equivalent to premises A and B.
>> Simply by assuming these premises, we are applying the topology of
>> pointwise convergence problem.
>
>Where? You did not get around to mentioning convergence at all until
>*after* you had completed the proof. That's a strange way of "applying"
>something.
Pointwise convergence is equivalent to premises A and B. By using the
premises, I was applying topology.
>By the way, your premises are specialized to "the end", but they can be
>restated to apply to any point in time in an interval where a marble is
>not being moved. I see no difference between "the end" and any other
>time when the generalized premises apply.
>
> [Lots of repetition snipped.]
Again, why add a lot of intervals in which nothing happens, and may
even confuse the issue.
>>>> This can
>>>> also be stated as: the function mapping the steps to the indicator
>>>> functions is continuous on the one point compactification of the
>>>> positive integers (which has one limit point, infinity).
>
>>>The indicator function f_n is continuous at the end, on the assumption
>>>that ball n is not moved at time t=t0 (the end). But the sum over all of
>>>the indicator functions is wildly discontinuous at the end.
>
>> I am confused as to what the sum over all the indicator functions has
>> to do with the problem.
>
>This was an aside, not directly related to anything you said, but it
>certainly is relevant to the original problem. The problem asks how many
>marbles are in the bag at the end, and the function I just described (f =
>sum_n f_n) gives precisely the cardinality of the set of marbles in the
>bag as a function of time. What the problem asks for is f(t0) = f(the
>end), and not lim_{t->t0-} f(t), which is quite different.
The sum over all the indicator functions would be the function that
tells the number of steps that each marble was in the bag. That is,
indexed by marble: {1,3,5,7,9,11,13,15,17,19,19,21,23,...}
What you seem to be talking about is the sum of the values of a single
indicator function, which tells the number of marbles in the bag after
a given step. That is, indexed by step: {10,9,19,18,28,27,37,36,...}.
>>>When you say an indicator function is continuous at t=t0 (the end), you
>>>are saying precisely that for each n:
>
>>> (1) lim_{t->t0-} f_n(t) exists,
>>> (2) f_n(t0) exists, and
>>> (3) f_n(t0) = lim_{t->t0-} f_n(t).
>
>>>But in making statements (2) and (3) you are agreeing with me that
>>>f_n(t0) has a meaning completely apart from the limit expression in (1).
>
>> I did not say that the function was continuous. The first part of what
>> I said was deleted in the quote above, but what I said was
>
>>|In any case, I can't talk about a value at the end, since that was
>>|never specified; the values were only specified at finite steps.
>>|Each marble is removed from the bag at some point and never returned.
>>|To say that that implies there are no marbles in the bag at the end,
>>|is to say that the indicator functions converge pointwise. This can
>>|also be stated as: the function mapping the steps to the indicator
>>|functions is continuous on the one point compactification of the
>>|positive integers (which has one limit point, infinity).
>
>Do you really mean to say that f(t0) = 0 iff (An) lim_{t->t0-} f_n(t)
>converges? I assume you meant to specify that each of the indicator
>functions converges to zero, but that's still not sufficient, because the
>indicator functions could be discontinuous at t0.
I apologize for any unclarity in my quote. Let me restate.
Statement C: Each marble is removed from the bag at some point and
never returned.
Statement D: The indicator functions of which marbles are in the bag
converge pointwise.
Statement E: The function mapping the steps to the indicator functions
is continuous on the one point compactification of the
positive integers
"Statement C implies there are no marbles in the bag at the end" is
equivalent to Statement D. Statement E is a more precise way to say
Statement D.
I did not mean to claim anything about whether Statement E was true.
However, if we claim that Statement C implies there are no marbles in
the bag, then Statement E is true.
>Your premises A and B prove to my satisfaction that the bag is empty at
>the end, but your own argument draws that conclusion without invoking
>limits, except by way of an addendum after the proof is complete.
Premises A and B are simply the premises for convergence under the
topology of pointwise convergence.
Rob Johnson <r...@trash.whim.org>
david petry
The anti-Cantorians don't think of Cantorians as
"crackpots", but rather as fanatical devotees of a
mythology.
Cantor's Mythology (it's not truly a "theory") is a
story about a world beyond what we can observe, and
it is inessential to the mathematics that helps us
understand the world in which we live.
david petry
David C. Ullrich wrote:
> On 25 Oct 2005 15:52:47 -0700, "david petry"
> <david_lawr...@yahoo.com> wrote:
> >In this marbles problem, things change one step at a time. So,
> >if every marble has been removed, there must be some step in
> >which the last marble was removed.
>
> No, that simply does not follow.
Why not?
> No. Instead I'll ask you _which_ marble is remaining at the end.
You refuse to admit that there are unstated rules involved
in your game. Other consistent games could be played.
Read what Rob Johnson has been writing on this.
david petry
Jesse F. Hughes wrote:
> "david petry" <david_lawr...@yahoo.com> writes:
> > When discussing these completed infinity kinds of things, it
> > is utterly absurd to try to use common sense, but ...
> >
> > In this marbles problem, things change one step at a time. So,
> > if every marble has been removed, there must be some step in
> > which the last marble was removed.
>
> What does "so" mean here?
"Common sense would suggest..."
Rob Johnson
David C. Ullrich <ull...@math.okstate.edu> wrote:
>On 25 Oct 2005 15:52:47 -0700, "david petry"
><david_lawr...@yahoo.com> wrote:
>
>>
>>David C. Ullrich wrote:
>>
>>> I don't see what "limits" in whatever sense have to do with it.
>>> Every marble that's added is also removed, so there are no marbles
>>> remaining.
>>
>>When discussing these completed infinity kinds of things, it
>>is utterly absurd to try to use common sense, but ...
>>
>>In this marbles problem, things change one step at a time. So,
>>if every marble has been removed, there must be some step in
>>which the last marble was removed.
>
>No, that simply does not follow.
>
>>But, we know that just before
>>that particular step, ten additional marbles were added, so we
>>must conclude that there are at least nine marbles still remaining.
>>With a little more analysis, we would find that there are even
>>more marbles remaining.
>
>We find that?
The point is that for a limit ordinal, there is no preceding ordinal.
We can't look at the previous step "at the end".
>>Ullrich will now point out that common sense is some silly kind
>>of thing which only cranks would rely on.
>
>No. Instead I'll ask you _which_ marble is remaining at the end.
>Let's give them names: say M1 is the first marble inserted,
>M2 is the second one inserted, etc.
>
>You say that we "find" that there are marbles remaining.
>So give us an example - is M1 remaining, is M2 remaining,
>etc?
Each given marble is taken out of the bag at some step, never to
return. This is pointwise convergence of the indicator functions to
the function indicating an empty bag.
Dave Seaman
> In article <djo519$ifs$1...@mailhub227.itcs.purdue.edu>,
> Dave Seaman <dse...@no.such.host> wrote:
>>> To end the process at noon, we can embed the index space (the positive
>>> integers) into [0,1] (e.g. by t = 1-1/n) and, since we wish to consider
>>> the value "at the end", map infinity in the one point compactification
>>> of the index space to 1 (noon). This leaves a lot of points in [0,1]
>>> with undefined values. We can fix that up by deciding that at a point,
>>> x, in [0,1) not covered by the embedded integers, the value is the same
>>> as at the greatest point which is covered and not greater than x.
>>> However, this does not tell us the value of the indicator function at
>>> infinity (or 1 or noon) any more than before.
>>While we are "fixing things up", why don't we just include "the end" in
>>the fixing process? What is so special about "the end"?
> We can't include "the end", because the state "at the end" can not be
> deduced from any of the preceding states unless we add some premises.
I will grant your premises A and B. I think we have already agreed that
they are sufficient to deduce the state of each marble at "the end". I,
however, would prefer to generalize those premises so that they apply to
other times besides "the end" at which a marble is not being moved. I
see no difference between "the end" and other times within the scope of
the problem.
> The problem tells how to go from one state to the succeeding state.
If marble n is added to the bag at time t, the indicator function f_n
jumps from 0 to 1 at time t. If marble n is removed, the indicator
function f_n jumps from 1 to 0 at time t.
> The problem says nothing about states which have no predecessor. To
> describe what happens at a state indexed by a limit ordinal, we must
> have more premises.
What do you mean by a state that has no predecessor? The indicator
functions f_n are defined for all times t.
> Furthermore, I don't see any reason to increase the domain from the
> positive integers (or possibly their one-point compactification) by
> adding a lot of real intervals on which nothing occurs. It may only
> confuse the issue.
My indicator functions are defined at least for the nonnegative reals
(possibly the one-point compactifiation of the nonnegative reals, or
possibly the long line, depending on where "the end" is).
If we don't add intervals on which nothing occurs, then your premises A
and B do not apply. That's precisely why they are needed. I see nothing
confusing about the statement that each f_n is constant on intervals
where marble n is not moved.
>>Let's simplify the problem by considering just a single ball. The ball
>>is added to an empty bag at 11:59. How many balls are in the bag at
>>noon? Do you claim the answer is unspecified, because noon is "the end"
>>and the problem doesn't specify what happens at "the end"?
>>What if the problem asks about 11:59:30 instead of noon? Does that mean
>>the answer is specified, or is it still unspecified because 11:59:30 has
>>now become "the end" and you have decreed that "the end" requires special
>>treatment?
>>Suppose there are two balls, one added at 11:58 and one at 11:59. How
>>many balls are in the bag at noon? How about 11:58:30? Do you claim
>>that the reasoning is somehow different for those two times, because one
>>is "the end" and the other is not?
> In these examples, the last state is indexed by a successor ordinal.
> The problem tells what happens in these cases. This is very different
> than talking about an unending sequence of ever changing states.
I thought we were talking about the original problem for this thread.
There is no n for which marble n has an unending sequence of ever
changing states. If there were such an n, then the state of the n-th
marble at "the end" would be undefined.
>>> A couple of additional premises can define, in special circumstances,
>>> which marbles are in the bag "at the end":
>>> A: if a marble is taken out of the bag at some finite step and never
>>> returned, then it is outside the bag "at the end"
>>> B: if a marble is placed into the bag at some finite step and never
>>> removed, then it is inside the bag "at the end"
>>I note that neither of your premises mentions limits or topology in any
>>way.
I will add that neither of your premises mentions anything more than a
single marble. What happens to marble n+1 has no relevance to the state
of marble n. We can solve the puzzle for any given n by removing from
consideration all marbles other than the n-th one, thus concluding that
marble n has only finitely many transitions and is not in the bag at the
end. Then we repeat the reasoning as n is allowed to range over the
naturals.
> "at the end" is infinity, the limit of the one-point compactification
> of the positive integers. These two premises describe the topology of
> pointwise convergence.
Understood. But you did not use that fact in your solution to the
puzzle. Pointwise convergence is merely a consequence of your reasoning,
not a prerequisite.
>>> These seem to be simply common sense when applied to physical models,
>>> but they are necessary to make the math reflect the physical model.
>>> These two additional premises imply that the bag is empty "at the end".
>>Still no mention of limits, and your proof is now complete. You have
>>just confirmed my premise that pointwise convergence is unnecessary to
>>the solution.
>>> However, these two additional premises are simply a restatement of
>>> what it means for the indicator functions of the marbles in the bag
>>> to converge pointwise. These two premises imply that the indicator
>>> functions converge pointwise.
>>I agree that pointwise convergence is a corollary of your argument, but
>>you have not used that corollary in the proof that you just presented.
> Pointwise convergence is equivalent to premises A and B.
Pointwise convergence is a consequence of premises A and B, assuming no
movement at "the end". You certainly can't claim equivalence unless you
at least take my suggestion to generalize A and B by letting them refer
to arbitrary times when the marble is not being moved.
A': if a marble is placed into the bag at time t=t1 and is
not removed at any time t in [t1,t2], then the marble
is in the bag at time t2.
B': if a marble is removed from the bag at time t=t1 and
is not replaced at any time t in [t1,t2], then the
marble is not in the bag at time t2.
Whether it is equivalent or not, the fact remains that you yourself
solved the puzzle without referring to pointwise convergence. This
contradicts your claim that convergence is necessary to the solution.
>>> Simply by assuming these premises, we are applying the topology of
>>> pointwise convergence problem.
No, because convergence fails if there is movement at "the end", but
premises A' and B' give a correct solution even in that case by taking t1
= t2 = "the end".
>>> I am confused as to what the sum over all the indicator functions has
>>> to do with the problem.
>>This was an aside, not directly related to anything you said, but it
>>certainly is relevant to the original problem. The problem asks how many
>>marbles are in the bag at the end, and the function I just described (f =
>>sum_n f_n) gives precisely the cardinality of the set of marbles in the
>>bag as a function of time. What the problem asks for is f(t0) = f(the
>>end), and not lim_{t->t0-} f(t), which is quite different.
> The sum over all the indicator functions would be the function that
> tells the number of steps that each marble was in the bag. That is,
> indexed by marble: {1,3,5,7,9,11,13,15,17,19,19,21,23,...}
We must be talking about different indicator functions. My indicator
functions are given by
f_n(t) = 1, if marble n is in the bag at time t,
= 0, otherwise.
It follows that f = sum_n f_n is the cardinality function, as I said.
The indicator functions f_n converge if there is no movement at "the
end", but the cardinality function f has an infinite discontinuity on the
left at "the end".
> What you seem to be talking about is the sum of the values of a single
> indicator function, which tells the number of marbles in the bag after
> a given step. That is, indexed by step: {10,9,19,18,28,27,37,36,...}.
No, I am talking about the sum of the values of all the indicator
functions, f(t) = sum_n f_n(t). That is the cardinality function.
Dave Seaman
> Why not?
Unless I have misunderstood things very badly, Rob Johnson has been
saying all along that the bag is empty at the end.
We may differ on the details of the argument, but I am reasonably certain
that the conclusion is not in dispute.
david petry
Dave Seaman wrote:
> On 26 Oct 2005 12:49:23 -0700, david petry wrote:
> > You refuse to admit that there are unstated rules involved
> > in your game. Other consistent games could be played.
>
> > Read what Rob Johnson has been writing on this.
>
> Unless I have misunderstood things very badly, Rob Johnson has been
> saying all along that the bag is empty at the end.
Yes, but he is spelling out what the rules of the game must
be to reach that conclusion.
Dave Seaman
As am I.
David R Tribble
>> What I really want to know is, Jiri: do you think that the jar is empty?
>> My bet is that you do.
>
Jiri Lebl wrote:
> Yes of course it is empty. The problem is that the statement of the
> problem makes it sound like something that could actually happen in
> real life. That's what gets all of you all riled up. If the problem
> (identical logical statement) would be stated in purely abstract terms
> without references to marbles, bags, jars or ping pong balls, nobody
> would get all that angry. It is a mathematical thought experiment, not
> a real life situation. Trying to apply either real life intuition or
> perhaps physical laws is bound to cause confusion. It is to
> demonstrate that you have to be careful in logical arguments involving
> large sets (in this case N).
Against my better judgement, I am posting to this n+1 thread dealing
with the same tired old puzzle.
Suppose we define a series of sets, beginning with set S(0):
S(0) = {}
S(i) = S(i-1) U {i+0, i+1, ..., i+9} \ {i}
for all i=1,2,3,...
In English, set S(i) is the previous set S(i-1) with elements i+0,
i+1, ..., i+9 added to it and with element i removed from it.
i is simply a counter, being a natural number starting at 0.
Now we define the union of all these sets:
Su = S(0) U S(1) U S(2) U ...
The question is, how many elements are in set Su?
ste...@nomail.com
More than you intended? :)
Stephen
Shmuel (Seymour J.) Metz
10/24/2005
at 09:41 PM, Star...@Earthlink.net said:
>By the nth iteration, n+1-n^2 will also be in the bag. So given we
>take the limit to infinity.
What do you mean by "take the limit to infinity"? Unless you define
it, and prove that it has whatever properties you are relying on, your
writings are meaningless.
>The infinitieth ball
What do you mean by "The infinitieth ball"? Unless you define it, and
prove that it has whatever properties you are relying on, your
writings are meaningless.
What do you mean by "infinity+1-10*infinity" and by "the
infinity+1-10*infinity ones"? Unless you define them, and prove that
they have whatever properties you are relying on, your writings are
meaningless.
>n approaches infinity, and 10n approaches infinity too
What do you mean by "n approaches infinity" and by "10n approaches
infinity"? Unless you define them, and prove that they have whatever
properties you are relying on, your writings are meaningless.
Copying words that you do not understand is not logic.
In <1130229559.6...@g47g2000cwa.googlegroups.com>, on
10/25/2005
at 01:39 AM, Star...@Earthlink.net said:
>Learn calculus!
PKB. You are extremely confused about what limits are in Calculus.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
David R Tribble
>> Against my better judgement, I am posting to this n+1 thread dealing
>> with the same tired old puzzle.
>>
>> Suppose we define a series of sets, beginning with set S(0):
>> S(0) = {}
>> S(i) = S(i-1) U {i+0, i+1, ..., i+9} \ {i}
>> for all i=1,2,3,...
>>
>> In English, set S(i) is the previous set S(i-1) with elements i+0,
>> i+1, ..., i+9 added to it and with element i removed from it.
>> i is simply a counter, being a natural number starting at 0.
>>
>> Now we define the union of all these sets:
>> Su = S(0) U S(1) U S(2) U ...
>>
>> The question is, how many elements are in set Su?
>
Stephen wrote:
> More than you intended? :)
Heh, yes. Good to know someone is paying attention.
I tried to avoid any explicit mention of "infinity" in that last step,
but it looks like it can't be avoided. Hmpfff.
Torkel Franzen
> The anti-Cantorians don't think of Cantorians as
> "crackpots", but rather as fanatical devotees of a
> mythology.
Of course. Similarly with anti-Einsteinians, anti-Darwinians, and so
on.
Dik T. Winter
> David R Tribble wrote:
...
> >> In English, set S(i) is the previous set S(i-1) with elements i+0,
> >> i+1, ..., i+9 added to it and with element i removed from it.
> >> i is simply a counter, being a natural number starting at 0.
> >>
> >> Now we define the union of all these sets:
> >> Su = S(0) U S(1) U S(2) U ...
> >>
> >> The question is, how many elements are in set Su?
> >
>
> Stephen wrote:
> > More than you intended? :)
>
> Heh, yes. Good to know someone is paying attention.
>
> I tried to avoid any explicit mention of "infinity" in that last step,
> but it looks like it can't be avoided. Hmpfff.
It is worse. For Su it does not matter whether elements are removed
when going from Si to Si+1. Su is a union.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Star...@earthlink.net
Star...@earthlink.net
Pubkeybreaker wrote:
> "Are you RETARDED? If you're taking this limit.... <snip"
>
> (1) Your comments and behavior make you look arrogant.
> (2) Your "math" consists of handwaving gibberish and shows great
> ignorance.
> (3) You use the combination to denigrate others.
>
> I will refrain from any perjoratives regarding how the combination of
> (1), (2), and (3)
> makes you look to outsiders.
>
> A countably infinite number of marbles was placed in the bag.
> A countably infinite number was removed.
>
> Since ALL countably infinite sets have the same size, there are no
> marbles
> left. Every labelled ball placed in the bag was removed at some time.
>
does NOT mean that infinity - infinity = 0.
10n approaches infinity, and n approaches infinity. They are both the
very same infinity. 10n - n is 9n and 9n approaches infinity as n
approaches infninity. You might arrange it in a way that makes it look
like it is zero, but then you are looking at the wrong angle of the
problem.
> Wait. Let me guess. You think that the set
>
> A = {0,1,2,3,4,5,6,7,8,9,10,11,12....}
>
> is "bigger" than the set
>
> B = {0, 10, 20, 30, 40, ...}
>
> Because the former contains elements that are not in the latter.
> Right?
>
> Well, I have news. Both sets are the same size. For each element in A
> there is
> one and only one corresponding element in B. And vice-versa.
Straw man.
Also, notice that A - B =
{1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,22,23,24 ...} and
that's the same size as A and B.
Shocked much?
Star...@earthlink.net
Shmuel (Seymour J.) Metz wrote:
> In <1130215291....@o13g2000cwo.googlegroups.com>, on
> 10/24/2005
> at 09:41 PM, Star...@Earthlink.net said:
>
> >By the nth iteration, n+1-n^2 will also be in the bag. So given we
> >take the limit to infinity.
>
> What do you mean by "take the limit to infinity"? Unless you define
> it, and prove that it has whatever properties you are relying on, your
> writings are meaningless.
>
every M there is an n that is higher than it.
> >The infinitieth ball
>
> What do you mean by "The infinitieth ball"? Unless you define it, and
> prove that it has whatever properties you are relying on, your
> writings are meaningless.
>
> What do you mean by "infinity+1-10*infinity" and by "the
> infinity+1-10*infinity ones"? Unless you define them, and prove that
> they have whatever properties you are relying on, your writings are
> meaningless.
>
Well I didn't exactly use mathematically correct terms, but my point
remains valid. What I meant was that ball n as n approaches infinity.
There's a reason why it's called infinity. There is no ONE NUMBER
infinity. Infinity encompasses a variety of different possible ways
that one can get to infinity. For example, 2n as n approaches infinity
is not the same number as n as n approaches infinity. Same with n
squared, the square root of n, and so on and so forth.
> >n approaches infinity, and 10n approaches infinity too
>
> What do you mean by "n approaches infinity" and by "10n approaches
> infinity"? Unless you define them, and prove that they have whatever
> properties you are relying on, your writings are meaningless.
>
Read above.
> Copying words that you do not understand is not logic.
>
I understand those words. I've just explained them to you.
>
> In <1130229559.6...@g47g2000cwa.googlegroups.com>, on
> 10/25/2005
> at 01:39 AM, Star...@Earthlink.net said:
>
> >Learn calculus!
>
> PKB. You are extremely confused about what limits are in Calculus.
>
I know exactly what they are. You just refuse to understand them,
perhaps because you failed calculus.
Star...@earthlink.net
one number called infinity? Even if we use only one TYPE of infinity,
there are still an infinite number of infinities.
We could always define infinity as a number for which N + 1 = N, but we
would get nonsense properties if we did that.
At the very least, this much can be said about the topic: Introducing
the concept of infinity is the same thing as introducing division by
zero. You get very badly behaved mathematics when you do this.
Star...@earthlink.net
Would the solution change?
We are always under the impression that in mathematics, no matter which
angle we use to identify the problem, the result will always be the
same. If this is not true, then we have a variety of different kinds of
mathematics all pertaining to different kinds of universes. If this is
true, then the simplest explaination ought to be taken to be the
correct one, a la Occam's Razor.
Virgil
Star...@Earthlink.net wrote:
> Pubkeybreaker wrote:
> > Every labelled ball placed in the bag was removed at some time.
> >
> That's a non sequitur. Just because both infinities are the same size
> does NOT mean that infinity - infinity = 0.
It isn't that, it is that at some time after each ball is put it, it is
then taken out and not put in again. Therefore there cannot be any
single ball that has not been removed.
Or does Starbles have in mind some ball which will remain in after being
taken out?
Star...@earthlink.net
> >A real line version of your marble problem would be like this:
> > What is the limit of the sequence of sets S_n = [n + 1,10n] ?
>
> Which could be answered using the process I gave above, in which case
> the answer would be that the limit is empty.
>
So you're saying that the set falls off the edge of the universe?
How do you even know that set theory is the right context for this
problem?
Set theory is not more correct than ordinary calculus just because it
is more advanced. It just describes a different kind of universe.
Any universe with ACTUAL infinities in it, BTW, is not a real univsere,
IMO.
Virgil
Star...@Earthlink.net wrote:
> > PKB. You are extremely confused about what limits are in Calculus.
> >
> I know exactly what they are. You just refuse to understand them,
> perhaps because you failed calculus.
As I have taught calculus successfully for years, I can say that you do
not seem to be quite clear about when "limits" can be said to exist.
There are infinite processes for which there is not anything that can
properly be called a limit to the process.
Virgil
Star...@Earthlink.net wrote:
> Just a question. Where in Cantor's set theory does it say that there is
> one number called infinity? Even if we use only one TYPE of infinity,
> there are still an infinite number of infinities.
If we are using distances, as in geometry, there is no distance greater
than an merely infinite distance. Cardinal or ordinal infinities are not
so limited.
>
> At the very least, this much can be said about the topic: Introducing
> the concept of infinity is the same thing as introducing division by
> zero. You get very badly behaved mathematics when you do this.
Cardinal and ordinal infinities are quite well behaved, if one allows
oneself to be guided by their inherent logic rather that any intuition
based on finiteness.
Star...@earthlink.net
equivocate that with emptiness, because you are engaging in context
dropping. The context that is dropped here is time.
Let's say that at t = I ball n has been put in. Then later, at t = J
the ball is removed. I != J. If I = J, then you could argue that there
are no balls. However, the process of adding and removing balls is
lopsided.
BTW, there is a difference between "Looking for a ball at spot n and
not finding it" and "The bag is empty". That is a BIG difference. The
balls that are still there are just unfindable. They are not missing.
You are apparantly engaging in equivocation.
I've pointed out your fallacies. It's time to fess up.
Star...@earthlink.net
every n. Thus at the nth iteration, there is no marble at spot n but
there is at n+1. There is no number for which n = n+1, since that
implies that 0 = 1.
Infinity is not a number. You can't REACH infinity. That's the whole
point. Saying "IF we reach infinity" is nonsense. That's like asking
"What if we reached the speed of light and turned our headlights on?"
or "What if we went to a time before the universe started". If the
universe is either an infinite amount of time old or a finite amount of
time old, that statement will be meaningless.
I will say this. If we can have a concept such as reaching infinity, we
can have a concept of a bag that gets so full that it is empty.
However, my argument is this: Both of these things do not exist in our
iniverse. Maybe in Cantor's universe, but Cantor's universe does not
describe reality, because reality is devoid of infinities. The concept
of infinity is just that: a concept.
Virgil
Star...@Earthlink.net wrote:
> > >A real line version of your marble problem would be like this:
> > > What is the limit of the sequence of sets S_n = [n + 1,10n] ?
> >
> > Which could be answered using the process I gave above, in which case
> > the answer would be that the limit is empty.
> >
> So you're saying that the set falls off the edge of the universe?
Only that there is no element common to all those sets, S_n.
>
> How do you even know that set theory is the right context for this
> problem?
One is being asked what elements are in a certain continer after an
cetrtain infinite process, presuming that that infinite process can be
completed in finite time, has been completed. This appears remarkably
like asking what is the membership of a some set.
>
> Set theory is not more correct than ordinary calculus just because it
> is more advanced. It just describes a different kind of universe.
Set theory results will be more correct than calculus in any situation
in which set theory is more appropriate than calculus.
A good deal of calculus involves continuous functions, even
differentiable functions, none of which appear in the ping pong ball
problem. And sequences or series in calculus have terms which are either
real numbers or real functions, not bunches of ping pong balls.
>
> Any universe with ACTUAL infinities in it, BTW, is not a real univsere,
> IMO.
It is certainly not physically realizable, but to humans, the world of
ideas is, in many senses, as "real" as the physical world.
Virgil
Star...@Earthlink.net wrote:
> At some later time, yes, it is taken out, but you cannot possibly
> equivocate that with emptiness, because you are engaging in context
> dropping. The context that is dropped here is time.
>
> Let's say that at t = I ball n has been put in. Then later, at t = J
> the ball is removed. I != J. If I = J, then you could argue that there
> are no balls. However, the process of adding and removing balls is
> lopsided.
It is sequential. Certainly no ball is removed BEFORE it is inserted,
but every ball is removed AFTER it is inserted, and by the time that
infinitely many stages of the entire process are completed.
>
> BTW, there is a difference between "Looking for a ball at spot n and
> not finding it" and "The bag is empty". That is a BIG difference. The
> balls that are still there are just unfindable. They are not missing.
> You are apparantly engaging in equivocation.
Looking for a ball and not finding it is also different from looking for
a ball and finding that it has been removed, which is the case for every
ball, at least by the time the entire process is complete.
>
> I've pointed out your fallacies. It's time to fess up.
I confess that Sarbles inability to understand is something I do not
understand. And that is all I have to confess.
Virgil
Star...@Earthlink.net wrote:
> If there is no limit to the process, then there is always an n+1 for
> every n. Thus at the nth iteration, there is no marble at spot n but
> there is at n+1. There is no number for which n = n+1, since that
> implies that 0 = 1.
>
> Infinity is not a number. You can't REACH infinity.
If that were really true then Zeno has proved trht a lot of things that
regularly do happen can't happen.
For example, suppose someone at point A wants to get to point B sme
disance away. That person must first go half the distance, but before
going half that distance, must go half of that half, and so on
infinitely.
So the person cannot even get started, much less arrive, unless that
person can take infinitely many steps in finite time.
> That's the whole
> point. Saying "IF we reach infinity" is nonsense.
Then so is motion.
> That's like asking
> "What if we reached the speed of light and turned our headlights on?"
> or "What if we went to a time before the universe started". If the
> universe is either an infinite amount of time old or a finite amount of
> time old, that statement will be meaningless.
Until you can solve all of Zeno's problems, you should not play around
with the notion of infinity, it will only confuse you.
>
> I will say this. If we can have a concept such as reaching infinity, we
> can have a concept of a bag that gets so full that it is empty.
> However, my argument is this: Both of these things do not exist in our
> iniverse. Maybe in Cantor's universe, but Cantor's universe does not
> describe reality, because reality is devoid of infinities. The concept
> of infinity is just that: a concept.
That is all that any of mathematics is, concepts.
But concepts rule in the real world.
Star...@earthlink.net
Virgil wrote:
> In article <1130388561.4...@f14g2000cwb.googlegroups.com>,
> Star...@Earthlink.net wrote:
>
> > If there is no limit to the process, then there is always an n+1 for
> > every n. Thus at the nth iteration, there is no marble at spot n but
> > there is at n+1. There is no number for which n = n+1, since that
> > implies that 0 = 1.
> >
> > Infinity is not a number. You can't REACH infinity.
>
> If that were really true then Zeno has proved trht a lot of things that
> regularly do happen can't happen.
>
> For example, suppose someone at point A wants to get to point B sme
> disance away. That person must first go half the distance, but before
> going half that distance, must go half of that half, and so on
> infinitely.
>
then walk half of that. We walk regularly. That is, when we walk, we
take strides.
> So the person cannot even get started, much less arrive, unless that
> person can take infinitely many steps in finite time.
>
They don't take infinitely many steps. Each step covers a certain
fininte distance.
>
> > That's the whole
> > point. Saying "IF we reach infinity" is nonsense.
>
> Then so is motion.
>
Nonsense.
>
> > That's like asking
> > "What if we reached the speed of light and turned our headlights on?"
> > or "What if we went to a time before the universe started". If the
> > universe is either an infinite amount of time old or a finite amount of
> > time old, that statement will be meaningless.
>
> Until you can solve all of Zeno's problems, you should not play around
> with the notion of infinity, it will only confuse you.
Zeno's problems, ALL of them, can be solved without treating infinity
as a real number.
> >
> > I will say this. If we can have a concept such as reaching infinity, we
> > can have a concept of a bag that gets so full that it is empty.
>
> > However, my argument is this: Both of these things do not exist in our
> > iniverse. Maybe in Cantor's universe, but Cantor's universe does not
> > describe reality, because reality is devoid of infinities. The concept
> > of infinity is just that: a concept.
>
> That is all that any of mathematics is, concepts.
>
> But concepts rule in the real world.
Not the concept of infinity. Infinity isn't really even a concept of a
number, because infinity isn't a number.
Star...@earthlink.net
Virgil wrote:
> In article <1130388064.1...@g44g2000cwa.googlegroups.com>,
> Star...@Earthlink.net wrote:
>
> > At some later time, yes, it is taken out, but you cannot possibly
> > equivocate that with emptiness, because you are engaging in context
> > dropping. The context that is dropped here is time.
> >
> > Let's say that at t = I ball n has been put in. Then later, at t = J
> > the ball is removed. I != J. If I = J, then you could argue that there
> > are no balls. However, the process of adding and removing balls is
> > lopsided.
>
> It is sequential. Certainly no ball is removed BEFORE it is inserted,
> but every ball is removed AFTER it is inserted, and by the time that
> infinitely many stages of the entire process are completed.
There is no such thing as completing an infinite number of steps.
> >
> > BTW, there is a difference between "Looking for a ball at spot n and
> > not finding it" and "The bag is empty". That is a BIG difference. The
> > balls that are still there are just unfindable. They are not missing.
> > You are apparantly engaging in equivocation.
>
> Looking for a ball and not finding it is also different from looking for
> a ball and finding that it has been removed, which is the case for every
> ball, at least by the time the entire process is complete.
Read above.
> >
> > I've pointed out your fallacies. It's time to fess up.
>
> I confess that Sarbles inability to understand is something I do not
> understand. And that is all I have to confess.
All you've done is prove that you don't understand how concepts work.
Russell
here's hoping my comments serve to quell rather than fuel
the mounting flames in this thread:
Virgil wrote:
> In article <1130384986....@z14g2000cwz.googlegroups.com>,
> Star...@Earthlink.net wrote:
>
> > Pubkeybreaker wrote:
>
> > > Every labelled ball placed in the bag was removed at some time.
But the relevant Pubkeybreaker quote (trimmed by Virgil)
was:
| Since ALL countably infinite sets have the same size, there are no
| marbles left.
| Every labelled ball placed in the bag was removed at some time.
The first sentence *is* a nonsequitur, as correctly
pointed out by mensanator in a later post, and also as
Starbles says here:
> > >
> > That's a non sequitur. Just because both infinities are the same size
> > does NOT mean that infinity - infinity = 0.
>
> It isn't that, it is that at some time after each ball is put it, it is
> then taken out and not put in again. Therefore there cannot be any
> single ball that has not been removed.
Right, this is the content of Pubkeybreaker's 2nd sentence,
which was correct.
>
> Or does Starbles have in mind some ball which will remain in after being
> taken out?
I've asked this myself, with no answer from Starbles.
I don't see know how Starbles *could* answer this; but I
notice he seems to be taking a different line of argument
now, along the lines that the whole scenario is impossible
since it requires an infinite number of steps to complete.
Of course in a practical sense he's right about that (for
the marble scenario) but Ullrich and others have made it
abundantly clear that we're not talking here about anything
practical. Btw I am wondering what Starbles's position is
on Achilles chasing the tortoise... is *that* an impossible
scenario in his opinion? (IMHO that would be an odd position
to take for someone who recommends studying CALCULUS.)
Virgil
Star...@Earthlink.net wrote:
> Virgil wrote:
> > In article <1130388064.1...@g44g2000cwa.googlegroups.com>,
> > Star...@Earthlink.net wrote:
> >
> > > At some later time, yes, it is taken out, but you cannot possibly
> > > equivocate that with emptiness, because you are engaging in context
> > > dropping. The context that is dropped here is time.
> > >
> > > Let's say that at t = I ball n has been put in. Then later, at t = J
> > > the ball is removed. I != J. If I = J, then you could argue that there
> > > are no balls. However, the process of adding and removing balls is
> > > lopsided.
> >
> > It is sequential. Certainly no ball is removed BEFORE it is inserted,
> > but every ball is removed AFTER it is inserted, and by the time that
> > infinitely many stages of the entire process are completed.
>
> There is no such thing as completing an infinite number of steps.
In order to move from point A to point B, one must first go half way,
then half the remaining way, then half of that remaining way, and so
on. So that if one cannot complete an infinite number of steps, one
cannot move at all.
>
> > >
> > > BTW, there is a difference between "Looking for a ball at spot n and
> > > not finding it" and "The bag is empty". That is a BIG difference. The
> > > balls that are still there are just unfindable. They are not missing.
> > > You are apparantly engaging in equivocation.
> >
> > Looking for a ball and not finding it is also different from looking for
> > a ball and finding that it has been removed, which is the case for every
> > ball, at least by the time the entire process is complete.
>
> Read above.
Those who don't complete infinitely many steps in finite time don't get
anywhere! Read above.
Russell
that some new additions have been made to the thread:
Russell wrote:
> I noticed what seems to be a small miscommunication here;
> here's hoping my comments serve to quell rather than fuel
> the mounting flames in this thread:
>
> Virgil wrote:
> > In article <1130384986....@z14g2000cwz.googlegroups.com>,
> > Star...@Earthlink.net wrote:
> >
> > > Pubkeybreaker wrote:
> >
> > > > Every labelled ball placed in the bag was removed at some time.
>
> But the relevant Pubkeybreaker quote (trimmed by Virgil)
> was:
>
> | Since ALL countably infinite sets have the same size, there are no
> | marbles left.
> | Every labelled ball placed in the bag was removed at some time.
>
> The first sentence *is* a nonsequitur, as correctly
> pointed out by mensanator in a later post, and also as
> Starbles says here:
>
> > > >
> > > That's a non sequitur. Just because both infinities are the same size
> > > does NOT mean that infinity - infinity = 0.
Hmm, I see that in later posts Starbles is now
claiming that infinities don't exist. But he didn't
object to using them (i.e. he didn't object specifically
to their *use*) in the above sentence. Evidently it is
OK to use them sometimes, e.g. when referring to the
size of sets? IMHO that would be an OK position, I'm
just trying to understand, not criticise here.
> >
> > It isn't that, it is that at some time after each ball is put it, it is
> > then taken out and not put in again. Therefore there cannot be any
> > single ball that has not been removed.
>
> Right, this is the content of Pubkeybreaker's 2nd sentence,
> which was correct.
>
> >
> > Or does Starbles have in mind some ball which will remain in after being
> > taken out?
>
> I've asked this myself, with no answer from Starbles.
> I don't see know how Starbles *could* answer this; but I
> notice he seems to be taking a different line of argument
> now, along the lines that the whole scenario is impossible
> since it requires an infinite number of steps to complete.
> Of course in a practical sense he's right about that (for
> the marble scenario) but Ullrich and others have made it
> abundantly clear that we're not talking here about anything
> practical. Btw I am wondering what Starbles's position is
> on Achilles chasing the tortoise... is *that* an impossible
> scenario in his opinion? (IMHO that would be an odd position
> to take for someone who recommends studying CALCULUS.)
Never mind, I see that Zeno has come up elsewhere in this
thread. IMHO Starbles has a non-mainstream take on Zeno
but it's not at all an unreasonable position. What *is*
unreasonable is for him to be so vehemently against (what I
am calling) mainstreamers who would say that you can complete
an infinite number of steps in finite time as long as the
time for each step gets shorter and shorter by a sufficient
amount.
Virgil
Star...@Earthlink.net wrote:
> Virgil wrote:
> > In article <1130388561.4...@f14g2000cwb.googlegroups.com>,
> > Star...@Earthlink.net wrote:
> >
> > > If there is no limit to the process, then there is always an n+1 for
> > > every n. Thus at the nth iteration, there is no marble at spot n but
> > > there is at n+1. There is no number for which n = n+1, since that
> > > implies that 0 = 1.
> > >
> > > Infinity is not a number. You can't REACH infinity.
> >
> > If that were really true then Zeno has proved trht a lot of things that
> > regularly do happen can't happen.
> >
> > For example, suppose someone at point A wants to get to point B sme
> > disance away. That person must first go half the distance, but before
> > going half that distance, must go half of that half, and so on
> > infinitely.
> >
> That's absurd. When we walk, we don't first walk half the distance,
> then walk half of that. We walk regularly. That is, when we walk, we
> take strides.
Does Starbles claim that as HE goes from any point A to any point B he
do NOT first cover half the distance between them, then cover the other
half? Or Does Stables move instantaneously from one point to another
without passing through the space between?
>
> > So the person cannot even get started, much less arrive, unless that
> > person can take infinitely many steps in finite time.
> >
> They don't take infinitely many steps. Each step covers a certain
> fininte distance.
But to get from point A to point B, does one not have to pass through
all the points in between? Are 'steps' discontinuous motions?
>
> >
> > > That's the whole
> > > point. Saying "IF we reach infinity" is nonsense.
> >
> > Then so is motion.
> >
> Nonsense.
No more so that yours.
>
> >
> > > That's like asking
> > > "What if we reached the speed of light and turned our headlights on?"
> > > or "What if we went to a time before the universe started". If the
> > > universe is either an infinite amount of time old or a finite amount of
> > > time old, that statement will be meaningless.
> >
> > Until you can solve all of Zeno's problems, you should not play around
> > with the notion of infinity, it will only confuse you.
>
> Zeno's problems, ALL of them, can be solved without treating infinity
> as a real number.
But not without treating real numbers as real numbers, and acknowledging
that any motion involves passage through an uncountable infinity of
intermediate positions.
>
> > >
> > > I will say this. If we can have a concept such as reaching infinity, we
> > > can have a concept of a bag that gets so full that it is empty.
> >
> > > However, my argument is this: Both of these things do not exist in our
> > > iniverse. Maybe in Cantor's universe, but Cantor's universe does not
> > > describe reality, because reality is devoid of infinities. The concept
> > > of infinity is just that: a concept.
> >
> > That is all that any of mathematics is, concepts.
> >
> > But concepts rule in the real world.
>
> Not the concept of infinity. Infinity isn't really even a concept of a
> number, because infinity isn't a number.
There are things other than numbers that are concepts and there are a
variety of concepts of infinity in mathematics. Calculus, for example,
cannot exist without some of them.
albs...@gmx.de
Jesse F. Hughes wrote:
>
> I don't really know the answer to this, but I strongly suspect that
> other mathematicians had used "actually" infinite sets long before
> Cantor. Why would people accept informal talk of infinitesimals but
> balk at infinite sets? It would be nice if someone knowledgeable
> about historical mathematics from Leibniz to Cantor could comment.
You don't know anything about this subject. But you have to say much
about it. Infinity was suspect from the beginning of contemplation of
men until the end of the 19th century when Cantor published his ideas.
In spite of this (or maybe due to that fact) math works with this
undefined idea of infinity before Cantor very successful. And it will
be fruitful again by the time "this disease (the idea of defined actual
infinity) will be terminated" (~ Poincare).
Maybe you are able to read some books about historical mathematics
(e.g. about or from Bolzano)?
>
> If I am right, then it is silly to call this position
> "anti-Cantorian".
You are wrong.
Regards
Albrecht Storz
albs...@gmx.de
Torkel Franzen wrote:
> Han de Bruijn <Han.de...@DTO.TUDelft.NL> writes:
>
> > Anti-Cantorian is a very good nomer.
>
> It signifies crackpottery.
No. It's a good indication for mental health.
Regards
AS
Russell
> Shmuel (Seymour J.) Metz wrote:
> > In <1130215291....@o13g2000cwo.googlegroups.com>, on
> > 10/24/2005
> > at 09:41 PM, Star...@Earthlink.net said:
[snip]
> > >The infinitieth ball
> >
> > What do you mean by "The infinitieth ball"?
[snip]
> Well I didn't exactly use mathematically correct terms, but my point
> remains valid. What I meant was that ball n as n approaches infinity.
Care to restate that? If n is changing, there is no
such thing as "that" ball.
Note, mathematicians do use the word "approach", but it's
shorthand for a definition we all know, that has nothing to
do with something moving. However, as you use it above,
you seem to have in mind some ball n that is moving toward
infinity... feel free to correct me, i.e. state what you mean
rigorously instead.
I don't think you'll be able to describe the "infinitieth
ball" rigorously, because we (i.e. we mainstreamers) know
there is no such thing. When we use infinities a la Cantor,
we are *not* talking at all about "infinitieth balls" or
such. Indeed, except for "approach" as I've noted, we
generally take considerable pains to avoid any locution
that even suggests such an interpretation. Look back at
what mathematicians have written in this thread and see if
that isn't the case.
In contrast, you are careless with your words. The upshot
is that now you now think you have to throw it *all* out.
Baby and bathwater alike.
[snip]
> > >Learn calculus!
> >
> > PKB. You are extremely confused about what limits are in Calculus.
> >
> I know exactly what they are. You just refuse to understand them,
> perhaps because you failed calculus.
It would help you to avoid embarrassment if you lurked a
bit and got to know some of the frequent posters in this
group, and their educational qualifications, better.
FWIW, I am not myself a mathematician, but I believe
Shmuel is (and so, could not have failed calculus). Btw
your harping on calculus suggests to me that this is the
highest you yourself have gone in math, am I right?
Jesse F. Hughes
> On Wed, 26 Oct 2005 18:03:18 GMT, Rob Johnson wrote:
>> In article <djo519$ifs$1...@mailhub227.itcs.purdue.edu>,
>> Dave Seaman <dse...@no.such.host> wrote:
>>>> To end the process at noon, we can embed the index space (the positive
>>>> integers) into [0,1] (e.g. by t = 1-1/n) and, since we wish to consider
>>>> the value "at the end", map infinity in the one point compactification
>>>> of the index space to 1 (noon). This leaves a lot of points in [0,1]
>>>> with undefined values. We can fix that up by deciding that at a point,
>>>> x, in [0,1) not covered by the embedded integers, the value is the same
>>>> as at the greatest point which is covered and not greater than x.
>>>> However, this does not tell us the value of the indicator function at
>>>> infinity (or 1 or noon) any more than before.
>
>>>While we are "fixing things up", why don't we just include "the end" in
>>>the fixing process? What is so special about "the end"?
>
>> We can't include "the end", because the state "at the end" can not be
>> deduced from any of the preceding states unless we add some premises.
>
> I will grant your premises A and B. I think we have already agreed that
> they are sufficient to deduce the state of each marble at "the end". I,
> however, would prefer to generalize those premises so that they apply to
> other times besides "the end" at which a marble is not being moved. I
> see no difference between "the end" and other times within the scope of
> the problem.
>
>> The problem tells how to go from one state to the succeeding state.
>
> If marble n is added to the bag at time t, the indicator function f_n
> jumps from 0 to 1 at time t. If marble n is removed, the indicator
> function f_n jumps from 1 to 0 at time t.
To make this slightly more precise, let g(n) be the time at which the
marble is added to the bag and h(n) the time at which it is removed.
By assumption, 0 <= g(n) <= h(n) < 1.
(Or 11:00 <= g(n) <= h(n) < 12:00 in the original statement.)
Then we see that
/ 0 if t < g(n)
f_n(t) = | 1 if g(n) <= t < h(n)
\ 0 if h(n) <= t
This seems like the only reasonable interpretation of the problem.
>> The problem says nothing about states which have no predecessor. To
>> describe what happens at a state indexed by a limit ordinal, we must
>> have more premises.
>
> What do you mean by a state that has no predecessor? The indicator
> functions f_n are defined for all times t.
Obviously.
The assumption that f_n(t) changes value only at g(n) and h(n) seems
uncontroversial. Surely, it is implicit in the statement of the
problem.
If there is anything odd at all here, it is the following two
assumptions:
(1) The actions of adding and removing balls are atomic, i.e., they
occur at some point in time rather than over some interval.
(2) An infinite number of operations can be performed in a finite
length of time.
The second assumption is necessary for the problem to make any sense
at all. The first is a fairly natural simplifying assumption and is
also more or less explicit in the problem statement.
--
Jesse F. Hughes
"What you call reasonable is suspect since you've proven yourself to
be an enemy of mathematics." -- James S. Harris defends the cause.
Han de Bruijn
> Han de Bruijn wrote:
>
>>[ ... Jiri's massive overkill argument deleted ... ]
>>
>>What I really want to know is, Jiri: do you think that the jar is empty?
>>My bet is that you do.
>
> Yes of course it is empty.
See? It's _that_ easy:
Definition
----------
A mathematician is a _Cantorian_ if and only if he finds that the jar
is empty / contains zero marbles, in the end.
> demonstrate that you have to be careful in logical arguments involving
> large sets (in this case N).
The problem is that I become tempted to think that mathematical _logic
as such_ is not even reliable. This has actually bothered me ever since
I learned about propositional logic at the university.
Consider, for example the following tautology:
(p => q) v (q => p)
Translated in English: (if the sun is shining then it is raining)
or (if it is raining then the sun is shining)
This statement is true in mathematical logic, but, in common speech
(English, Dutch or whatever language) it is obviously false. Oh, now
don't explain me that common speech so unreliable when compared with
mathematical logic ! In my view, formal proofs are only acceptable if
they have a counterpart in common speech that "sounds good" at least.
Han de Bruijn
Han de Bruijn
> Unless I have misunderstood things very badly, Rob Johnson has been
> saying all along that the bag is empty at the end.
>
> We may differ on the details of the argument, but I am reasonably certain
> that the conclusion is not in dispute.
This classifies both Dave Seaman and Rob Johnson as Cantorians.
Just to keep the record up-to-date.
Han de Bruijn
Han de Bruijn
> "david petry" <david_lawr...@yahoo.com> writes:
>
>>The anti-Cantorians don't think of Cantorians as
>>"crackpots", but rather as fanatical devotees of a
>>mythology.
>
> Of course. Similarly with anti-Einsteinians, anti-Darwinians,
> and so on.
Of course, why not mix up everything, huh ?
Like atheists, socialists, neo-positivists, pink objectivists.
Han de Bruijn
Han de Bruijn
Hey! Seems that we are getting somewhere! That's a _very_ useful notion.
I repeat it in my own words:
In mathematics, no matter which route you take to arrive at a solution,
the result will always be the same.
The debate in threads like this indicates that something quite harmful
has happened to mathematics.It seems that the conclusion achieved is no
longer independent of the way you arrive at it. Something must be wrong
with the pathways of our reasoning. And no, calling each other names is
not enough of an argument. It never was.
Han de Bruijn
Han de Bruijn
> It isn't that, it is that at some time after each ball is put it, it is
> then taken out and not put in again. Therefore there cannot be any
> single ball that has not been removed.
Herewith Virgil is classified as a Cantorian too. Just for the record.
Han de Bruijn
David Kastrup
Just for the record: I don't think anybody cares about this Bruijnian
motion. Yawn.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
T.H. Ray
> mag...@math.berkeley.edu
> (Arturo Magidin) wrote:
>
> >In article
> <djevjl$7i9$2...@mailhub227.itcs.purdue.edu>,
> >Dave Seaman <dse...@no.such.host> wrote:
> >>On Sat, 22 Oct 2005 22:39:04 +0000 (UTC), Arturo
> Magidin wrote:
> >>> In article
> <39kkl11tcdhigqin4...@4ax.com>,
> >>> David C. Ullrich <ull...@math.okstate.edu>
> wrote:
> >>>>On Fri, 21 Oct 2005 16:32:03 +0000 (UTC),
> mag...@math.berkeley.edu
> >>>>(Arturo Magidin) wrote:
> >>>>
> >>>>>In article <tz86f.10494$cg....@news02.roc.ny>,
> >>>>>Steven <sgott...@hotmail.com> wrote:
> >>>>>>
> >>>>>> My friend and I have have different answers to
> the following problem. Who
> >>>>>>is correct and why?
> >>>>>>
> >>>>>>You have a countable number of marbles and a
> really big bag.
> >>>>>>The marbles are labeled 1, 2, 3, ... and so on.
> >>>>>>You put the first ten marbles into the
> aforementioned bag and then you take
> >>>>>>marble number 1
> >>>>>>out of the bag and discard it. Then you put
> marbles 11 through 20 into the
> >>>>>>bag and then you
> >>>>>>take marble number 2 out of the bag and discard
> it, and so on.
> >>>>>>In the end how many marbles are in the bag?
> >>>>>>
> >>>>>>
> >>>>>>Friend's Answer: This process does not have a
> well defined limit.
> >>>>>
> >>>>>Depends on your definition of limit.
> >>>>
> >>>>I don't see what "limits" in whatever sense have
> to do with it.
> >>
> >>> I agree... up to a point.
> >>
> >>>>Every marble that's added is also removed, so
> there are no marbles
> >>>>remaining.
> >>
> >>> Yes: if you go on long enough, you will reach a
> point where any
> >>> specific marble is taken out.
> >>
> >>> But if you want to talk about what happens "in
> the end", "and so on", with
> >>> an infinite process, surely some sort of limiting
> process is needed to
> >>> discuss this "end state".
> >>
> >>That only makes things more difficult. To solve
> the problem directly
> >>(without limits), you only need to answer one
> question: which balls are
> >>left "in the end"?
> >
> >What does "in the end" mean in a process that has no
> end?
>
David Ullrich replied:
> It means "after all the infinitely many steps have
> been carried
> out".<<
There are no infinitely many steps for a bag that
was defined to be finite but unbounded -- in this
context, an inexhaustible space.
However much this problem has been discussed and debated,
I find Arturo Magidin's explanation is still the most
rational. It relies on a necessary context.
One cannot speak of states that involve the counting of
discrete objects without reference to the limit of the
counting process, which is itself a discrete state.
We would otherwise be compelled to introduce mystical
concepts in which infinity is operational, instead of
admitting (as in this context)"no well defined limit."
So the discrete states that are tractable to definition
in this problem reduce to "empty" or "not well defined."
In fact, we know that in the physical world, quantum
states analagously relate to a counting process where
the energy level is measurably known ("empty" state) or
unknown ("not well defined"). One would get in trouble
confusing the unbounded bag (quantum phase space) with
the quantum number (measured energy).
One of the most marvelous aspects of mathematics, IMO,
is that while we creatively use the concepts "limit" and
"function," we still don't know much about them, or
even how or if "time" discretely separates defined
states in the limit. This is the problem that Einstein
ran up against in trying to reconcile continuous functions
with quantum uncertainty.
>
> Which of course is impossible in the real world. But
> saying
> something about limits does not change the fact that
> in
> the real world we can only do finitely many things
> before
> we die - the problem _is_ an abstract mathematical
> thing
> regardless.
>
> The state of some system "after infinitely many
> steps"
> need not always be well-defined. For example, you
> toggle
> a light switch on and off infinitely many times;
> there
> is no answer to the question of whether it's on or
> off at the end. But that's not really because a
> certain
> limit does not exist:
>
> Q: Start with x = 1. At each step replace the value
> of x with the current value of x divided by 2.
> What is the value of x after infinitely many
> steps?
>
> If someone said the answer was 0 because
> the limit of x was 0 I wouldn't feel inclined
> to argue with that. But if someone said that
> for _this_ problem "the value of x after infinitely
> many steps" was undefined I wouldn't be inclined
> to argue with _that_ either! Because the limit
> of x_n is _not_ the same thing as "the value
> of x after saying n = n+1, x = x_n infinitely
> many times"...
>
> In the current problem one could say that a certain
> sequence of sets has an empty limit. But that
> doesn't seem to me to be exactly what the problem
> is asking, and the answer to exactly what the
> problem is asking _is_ clear, without mentioning
> limits: Every marble that is added is also removed
> at some later stage, so after infinitely many
> steps the jar is empty.
>
> >How does the
> >question you ask me to answer even begin to make
> sense without
> >addressing that point? To solve the problem
> "directly" as you suggest
> >requires you to interpret what "the end" means for a
> process with NO
> >end. You may be hiding it, but you are still
> considering limits of
> >some kind at the very instant you start talking
> about "in the end". <<
That seems mystical me. "Hiding" it? There is quite some
semantic difference between hiding a fact and knowing
-- i.e., being able to define -- what it is.
> >
> >That said, since for most people it is intuitively
> clear what "in the
> >end" means for the purpose of this mental exercise,
> then for most
> >people it is indeed completely unnecessary (and only
> more confusing)
> >to introduce limits. Simply pointing out that ball n
> will be removed
> >in step n is enough to convince most people that
> there will be no ball
> >"in the end" (whatever that means).
> >
> >>If you introduce limits, you have to answer at
> least 3 questions:
> >>
> >> (1) What sort of limit should we consider?
> >> (2) What sort of value do we get for this limit?
> >> (3) Does this limit agree with the answer that we
> would have gotten
> >> without applying limits? That is, is the
> function in some sense
> >> continuous "at the end"?<<
"In some sense?" It is either in a discrete definable
state, or it is continuous -- no?
Tom
> >>
> >>It seems to me that in order to answer question
> (3), you have to answer
> >>the very same question as before, namely, what
> answer do you get without
> >>applying limits at all? And if you fail to
> consider question (3), then
> >>you simply have not answered the question that was
> asked.
> >
> >I did not bring up limits to answer the question, in
> any case. The
> >original poster said his friend answered that there
> was "no answer"
> >because "the" (singular definitive article) limit of
> the sets in
> >question did not exist. I pointed out that even if
> you want to try to
> >interpret the question by invoking limits
> explicitly, his friend's
> >analysis was flawed because the kind of limits he
> was considering was
> >far too limited for the task at hand. If ->he<-
> really wants to
> >consider limits, he has to address your question
> (1), at which point I
> >pointed out a reasonable extension of ->his<- notion
> of limit that
> >would render ->his<- objection moot.
>
>
> ************************
>
> David C. Ullrich
Han de Bruijn
> How do you even know that set theory is the right context for this
> problem?
Another breakthrough in thinking about mathematics: the question which
CONTEXT is appropriate for the problem at hand. That question would not
be relevant if mathematics could be considered as a whole, unified with
itself. And unified as well with the (other) sciences.
> Set theory is not more correct than ordinary calculus just because it
> is more advanced. It just describes a different kind of universe.
Well, it _should_ describe the same kind of universe, but, in practice,
it does not. And BTW, "ordinary" calculus is the more advanced, not set
theory.
> Any universe with ACTUAL infinities in it, BTW, is not a real univsere,
> IMO.
Affirmative.
Han de Bruijn
T.H. Ray
Have you considered Dedekind Cuts or the Cauchy Limit? You might find that they preserve your logical order
without contradiction. Quite constructive and formal.
Tom