Thinking Badly About Infinity II
Math1723
"sequel" for those wishing to continue the discussion. There is
already one dedicated to Tony Orlow's point of view on the matter,
which may be the one you want to go to. As the OP of the original
thread, I offer this sequel for those wishing to discuss it further in
a more general way (remembering that TO's discussion will likely
continue in his thread).
ORIGINAL POST:
-------------------------------------------------
This is usually given in the form of a puzzle or brain teaser, but I
want to present it more as a Math lesson, and as a detector for people
to see how critically well they think Mathematically, even when their
intuition goes contrary to logical dictates. (Often, cranks will
abandon logic to embrace the emotion, irrespective of the logic.) Many
of you know this example already as "The Bucket & Balls Problem".
We have an infinite number of Balls, all uniquely labelled #1, #2,
etc. A Clock is counts down from 1 second:
At time t=1 sec, Balls #1 thru #9.
At time t=1/2 sec, Ball #1 is removed, and Balls #10-19 are added.
At time t=1/4 sec, Ball #2 is removed, and Balls #20-29 are added.
At time t=1/8 sec, Ball #3 is removed, and Balls #30-39 are added.
...
At time t=1/2^n, Ball #n is removed, and Balls #10n thru 10n+9 are
added.
The clock and the process all stop at time t=0 seconds.
Note that by rule, no ball is added unless it is labelled with a
finite natural number. Repeat: No ball without a finite natural
number label is allowed to go in to the bucket. (You can pretend
there is a filter checking for the number label, like a bar code scan,
rejecting all balls lacking the appropriate label.)
Question: How many balls are in the bucket when the clock stops at
t=0?
Obviously, the number of balls in the bucket gets larger and larger as
the clock winds down. The number of balls at each time t=1/2^n
increases exponentially. The limit tends to infinity. But remember,
not all functions evaluate at their limit.
As it turns out: there are 0 balls in the bucket. We can know this
since for every Ball #n, we know that it was added at time t=1/2^[n/
10] and removed at time t= 1/2^n. We also know this must be true for
every single finite n (the only n's that were allowed in the bucket).
Put another way, ANY ball that is added at time t=1/2^x MUST BE
removed no later than time t=1/2^10x.
Now the logic is irrefutable. It may be counter-intuitive, defies
imagination, and seem downright wrong. But the logic is absolutely
sound, and if you accept logic, you must accept this. All the balls
are gone.
If you do not, you must ask yourself: If any balls exist in the
bucket, what numbers do they contain? They must be finite numbers (by
rule). Yet how could they have not been removed? Since every non-
empty set of finite natural numbers has a least element, then if any
balls exists in the bucket, there is one with the smallest number.
What is that number?
The purpose of this is to show how easily it is for us to use our
"intuition" and our "common sense" reasoning to think incorrectly
about infinities. Our experience in the finite world gives us
absolutely no handle on how things operate at transfinite levels. We
must rely on the logical rigors of Mathematics to answer our
questions, even if those rigors give us answers we don't like.
Thanks for reading.
Math1723
> On Jan 21, 5:24 pm, Tony Orlow <t...@lightlink.com> wrote:
> > But no ball is removed unless 10 are added
> For every ball removed, there are 10 balls added. However,
> it is also the case that for every ball added, there are 10
> balls removed. Isn't infinity fun?
Quite right! Well played, Marshall! :-)
Math1723
> On Jan 21, 1:01 pm, Math1723 <anonym1...@aol.com> wrote:
> > > I know that I've brought this up in different threads. I
> > > believe that in such a scenario, the "crank" role would be
> > > reversed, so that those who post something like standard
> > > ZFC would be called "cranks" instead.
> > Hell, if any of the "cranks" here simply justified their statement
> > based upon some other equi-consistent Set Theory, this wouldn't be a
> > problem! They wouldn't be cranks!
> Which is the main reason I try to look for that equi-consistent
> set theory -- so that they wouldn't be "cranks" anymore. But
> unfortunately, finding equi-consistent set theories that
> satisfying their desiderata is seldom _easy_ -- which is why I
> haven't been very succesful at doing so _yet_.
It would certainly be challenging. The main problem is that each ball
added is removed at some time t=1/2^n. To argue that the limit set is
non-empty, you have to identify at least one item, and state when it
was added. I'm not saying there isn't a consistent set theory of some
kind that would manage it (I'm not a Set Theoretician, so I am not
educated enough to say), but it would be hard to imagine that one
would be any less counter-intuitive than the problem you are trying to
"fix".
> > Cranks aren't cranks because they prefer another Set Theory. They are
> > cranks because they simply are unwilling (or unable) to follow basic
> > mathematical reasoning. Given starting assumptions, you get certain
> > conclusions. Different assumptions bring different conclusions.
> > Mathematicians are fine with that.
> I'm fine with that too! Given certain assumptions, we get the
> conclusion that the vase is empty. Given _other_ assumptions,
> we conclude that the vase is full. And all I'm trying to do
> is find the assumptions -- the _axioms_ -- from which we can
> validly conclude that the vase is full.
Therein lies the rub!
> > Over and over again, you hear me ask (ad naseum) to name a Ball that
> > is left in the vase at the end.
> Note that it's remotely possible -- depending once again on
> the axioms -- that one can prove the vase to be nonempty, but
> one can't actually name (construct) any ball in the vase.
This is an EXTREMELY good point, one which I have been waiting to hear
someone mention! It would need to be what is called an omega-
inconsistent theory (see: http://en.wikipedia.org/wiki/%CE%A9-consistent_theory
) I am not an expert in Mathematical Logic (of which this is a
branch), so I can't speak any more eloquently about it than you find
in Wikipedia or other places, but it is a fascinating study.
For me though, it is counter-intuitive to have a theory in which a
property of the natural numbers is individually false for 0, 1,
2, ..., n, ..., and false for every natural number, but the statement
"There exists a natural number with this property" is a theorem.
Certainly more counter-intuitive to me than the Bucket & Ball
problem. But that of course is just my opinion.
> After all, in ZFC, the set of all wellorders of R is also
> nonempty, but one can't actually name (construct) any
> wellorder of R. The proof that the set of wellorders of R
> is nonempty is _nonconstructive_ -- and it could be that
> the same is true for the vase (in the alternate theory).
The reason for this is that the Well Ordering Theorem is equivalent to
the Axiom of Choice, which itself is a non-constructive axiom. If it
could be constructive, the Well ordering would be a theorem, which in
turn would the Axiom of Choice, and ZF+~AC would become provably
inconsistent.
By the way, there are some models in which it is provably true. In
the Constructible universe, AC (and Well Ordering) are in fact
theorems. I am not suggesting the Axiom of Constructibility is one
which would be of interest to you, but just so that you know that AC
is not necessarily an absurd idea.
Math1723
> Well, you can't say that epsilon > 0 either.
> See John L Bell "Infinitesimals and the Continuum"http://publish.uwo.ca/~jbell/New%20lecture%20on%20infinitesimals.pdf
> page 10, where delta is the set of nilsquare infinitesimals:
> "The members of delta are all simultaneously <= 0 and >= 0, but cannot
> be shown to coincide with zero"
> and later: "we cannot decide of two 'events' in delta
> which came first; in fact it makes the stronger assertion that
> actually
> neither comes 'first'"
This was always the "cognitive friction" I had with SIA. With the Law
of the Excluded Middle, you can always know that exactly one of e<0,
e=0 or e>0 was true. Without LEM, you cannot.
Math1723
> Time to look at another version of the problem. The balls are now all
> labeled with the natural numbers 1, 2, 3, 4, ... as before, but now in
> an invisible ink that only mathematicians can see. New scenarios:
> Scenario 1) I begin by adding balls 1 through 10, and remove ball 1.
> At the next step, add balls 11 through 20, remove ball 3. Then, add
> balls 21 through 30, remove ball 5. Repeat. This leaves a countable
> infinity of balls in the urn. (Balls numbered 2, 4, 6, 8...)
Correct.
> Scenario 2) Same as above, but remove ball 2 at the first step, ...,
> remove ball n+1 at the n-th step. This leaves 1 ball in the urn. (The
> ball numbered 1.)
Correct again.
> Scenario k) Make up your own version!
> The procedure can be adjusted to leave any countable (finite or
> countably infinite) number of balls in the urn. To an observer who
> can't see the ink, all of the scenarios look exactly alike.
Seeing the ink is not the operative issue! This issue is that (in the
original problem) the Ball added at time t=1/2^[n/10] is removed at
t=1/2^n. The label is irrelevant; it is merely a convenience so we
can describe which ball is which.
The error is to assume it doesn't matter which ball is removed. It
makes all the difference in the world.
Key point: For each ball added, at what time (if any) is it removed?
If it is removed, then obviously it cannot be in the limit set. If it
is not removed, then obviously it remains in the limit set. You look
at this on a Ball-for-Ball basis.
It REALLY IS as simple as that!
If it helps at all, you can modify the problem so that a ball is
chosen at random to be removed. In this case, it is possible to have
a finite or zero sized bin at the limit, but the measure of this
probability is 0. In other words, you have to choose wisely to get
the result of the original problem (hence why I chose it).
Math1723
> You present a question with no answer: which natural number is left?
The question does indeed have an answer: No natural number is left, so
therefore the bin is empty. Once you see this, my question becomes a
rhetorical one.
> I also present an unanswerable question: when do the balls disappear?
This is also answerable question: They each disappear at different
times, namely Ball #n disappears at t=1/2^n. Only one ball at a time
disappears, so there is never a point at which "balls" disappear. If
you wish to look at it collectively though, they are all removed by
t=0.
> If you do not like that question, then you have stated the question
> with a dimension unfitting your position, and should leave time out
> of it, or at least let it take it's normal course and not pretend to
> end an unending sequence.
This issue has nothing to do with how I like or dislike a question.
The issue is what logic dictates. Logic requires a non-empty bin to
contain a ball with a label containing a finite natural number. Logic
does not require there to be a final ball to be removed for the
process to end at t=0.
> Personal question: Do you consider me a crank? I accept that title
> only because I have a beef with the current theory and refuse to
> concede that there is no better approach, but I do not accept your
> characterization as someone who won't listen to logic, if that's what
> you think.
Well, when answering questions, I try to take everything at face
value, without an assumption of any individual's crankhood. I feel
that is the most respectful way to handle it. Having said that, that
doesn't mean I am blind to the cranks that abound here on sci.math.
Consider JSH for example (I presume you are familiar with him). Years
and years have gone by with his trying to solve FLT using basic
Mathematical approaches, and many people here attempt to show him any
errors in his arguments (where they are coherent; tend to ignore the
incoherence).
None of the questions you have asked are necessarily crankish. Asking
about a "last integer", and other thought experiments of the kind, are
ones many students of Mathematics have explored early on. Where you
exhibit crankish *behavior* is your continual posting of the same
questions (despite knowing what the answer is already). You also seem
to approach Mathematics and Logic as if they were matters of opinion.
They are not. A theorem is not an opinion, it is a statement of
truth.
Now admittedly, every theorem begins with basic axioms (both
mathematical and logical), and sometimes therein lies the difference
of opinion. A Constructivist does not accept the Law of the Excluded
Middle, and therefore what may appear to be cranky is merely different
assumptions in your starting points. Non-Euclidean Geometers might
appear to be cranks to those who don't realize that Euclid's Parallel
Postulate is not being assumed.
But I have not heard you try to articulate a logical framework for
your position. I would expect a non-crank to proceed this way:
"I agree that with the current axioms of Mathematics and Logic, that
the bin is empty of balls. However, I believe you can change the
answer to a non-empty bin by changing the following axioms: ... "
Although I have seen you take pot-shots at ZFC (the standard axiomatic
system of Set Theory), I believe your results would require more
fundamental changes than simply at the Set-Theoretic level. Frankly,
I doubt you could create a consistent system that would do it. (By
the way, if you don't require consistency in your system, then simply
add "0=1" to your axiom set, and any result you like will be a
theorem.)
I feel pretty guiltless is categorizing JSH, Archimedes Plutonium, and
a few others here as cranks. However, the fact that you appear to be
sincerely engaged in dialog here (at least with me), makes me
unwilling to apply such a label to you (at least this time). Sadly
though, there are many others here, whose opinions I greatly respect,
and who perhaps know you better than I, feel no such hesitation in
including you in that lot. Perhaps that should be a bigger worry, and
perhaps it is they to whom you should have this discussion.
Good Luck.
Ray Vickson
In a previous post about this I posited the following scenario (based
on a Wiki article): to drive yourself crazy, look at the modification
in which at t = 1 we put into the urn balls 1--9; at t = 1/2 sec we
add balls 10--19 and remove ball 9; at t = 1/4 we add balls 20--29 and
remove ball 19, etc. NOW there are plenty of balls at t = 0. Aside
from re-numbering, the exact same add/remove operations are performed
in each case!
R.G. Vickson
Marshall
Thanks! I've made this point a few times, and none
of the cranks has ever addressed it that I saw. The
fact that there is a bijection between N and its infinite
subsets doesn't seem to be something that's at all
palatable to some people.
Which I suppose I have some sympathy for. It freaks
me out that the rational number are countable, for
example. But when presented with such a bijection,
which facet of one's experience does one attend to
more, the freakout or the knowledge of the bijection?
What does one conclude: infinite sets have surprising
properties, or modern math is a vast conspiracy of
error?
We should probably not be at all surprised that
infinite sets are surprising.
Marshall
Math1723
>
> In a previous post about this I posited the following scenario (based
> on a Wiki article): to drive yourself crazy, look at the modification
> in which at t = 1 we put into the urn balls 1--9; at t = 1/2 sec we
> add balls 10--19 and remove ball 9; at t = 1/4 we add balls 20--29 and
> remove ball 19, etc. NOW there are plenty of balls at t = 0. Aside
> from re-numbering, the exact same add/remove operations are performed
> in each case!
Yes, in this case the bin contains an infinite number of balls at the
end. You can modify the puzzle so that there are any finite number of
balls left as well.
The operative issue (in the original problem) is that for every n,
Ball #n is added at time t=1/2^[n/10] and is removed at t=1/2^n. That
is to say, every ball that got added was removed.
In your modified example, that's not true. Ball #1 was added at t=1,
but was never removed. We see that Ball #10n-1 is added at time
t=1/2^n and removed at t=1/2^n+1, so only Balls with labels of the
form 10n-1 are missing from the limit set.
The error is to assume it doesn't matter which ball is removed. It
makes all the difference in the world!
William Hughes
Yep, You add a subset of cardinality aleph_0
and remove a subset of cardinality aleph_0. This is not
enough information to allow you to decide on the cardinality
of the subset that remains. So if you change which numbers
you remove you change which subset you remove and the answer
changes. You do not change the cardinality of the subset you
remove so you do not have to change the number of balls you
remove at each step.
- William Hughes
Arthur
> On Jan 25, 3:02 pm, Arthur <arthur.guet...@zoho.com> wrote:
>
> > Scenario k) Make up your own version!
> > The procedure can be adjusted to leave any countable (finite or
> > countably infinite) number of balls in the urn. To an observer who
> > can't see the ink, all of the scenarios look exactly alike.
>
> Seeing the ink is not the operative issue! This issue is that (in the
> original problem) the Ball added at time t=1/2^[n/10] is removed at
> t=1/2^n. The label is irrelevant; it is merely a convenience so we
> can describe which ball is which.
>
> The error is to assume it doesn't matter which ball is removed. It
> makes all the difference in the world.
What error did I make? Where did I "assume"? You and I agree
completely on the "key point" that follows:
> Key point: For each ball added, at what time (if any) is it removed?
> If it is removed, then obviously it cannot be in the limit set. If it
> is not removed, then obviously it remains in the limit set. You look
> at this on a Ball-for-Ball basis.
>
> It REALLY IS as simple as that!
Yes, I agree it is as simple as that -- why did you read my post in
any other way?
Math1723
>
> What error did I make? Where did I "assume"? You and I agree
> completely on the "key point" that follows:
>
> > Key point: For each ball added, at what time (if any) is it
> > removed?
> > If it is removed, then obviously it cannot be in the limit set.
> > If it is not removed, then obviously it remains in the limit set.
> > You look at this on a Ball-for-Ball basis.
>
> > It REALLY IS as simple as that!
>
> Yes, I agree it is as simple as that -- why did you read my post in
> any other way?
Hi Arthur, sorry I did misread your post. Rereading your post again,
I see that I had injected "the error" with my own assumptions. My
apologies. (I hate it when people do that with me, and here I go do
it to someone else.)
Thanks for making the point and bringing it out!
David R Tribble
> For every ball removed, there are 10 balls added. However,
> it is also the case that for every ball added, there are 10
> balls removed. Isn't infinity fun?
Even more fun than that.
For every ball added, there are an infinite number of balls
removed. Specifically, after ball k is inserted, all of the
balls with labels that are integral multiples of k are removed
at some later time (one at a time, of course, and provided
that they were not already removed earlier).
David R Tribble
> In a previous post about this I posited the following scenario (based
> on a Wiki article): to drive yourself crazy, look at the modification
> in which at t = 1 we put into the urn balls 1--9; at t = 1/2 sec we
> add balls 10--19 and remove ball 9; at t = 1/4 we add balls 20--29 and
> remove ball 19, etc. NOW there are plenty of balls at t = 0. Aside
> from re-numbering, the exact same add/remove operations are performed
> in each case!
Well, exactly the same _number_ of add/remove operations
are performed, but a _different_ set of balls are removed than
in the original problem. And a completely different set of balls
are never removed from the vase.
So yours is a completely different problem.
Ray Vickson
Yes, I understand that. It just seems strange that what is left in the
urn (if anything) should depend on the numbering---although, let me
repeat---I agree that it does. Why can't we just add and subtract un-
labeled balls? Of course, in a sense they are still numbered in
principle, as first, second, third, etc. But what prevents us from
becoming memory-impaired during the procedure, and thus start removing
the "wrong" balls by mistake?
R.G. Vickson
Math1723
>
> Yes, I understand that. It just seems strange that what is left in the
> urn (if anything) should depend on the numbering---although, let me
> repeat---I agree that it does. Why can't we just add and subtract un-
> labeled balls? Of course, in a sense they are still numbered in
> principle, as first, second, third, etc. But what prevents us from
> becoming memory-impaired during the procedure, and thus start removing
> the "wrong" balls by mistake?
The operative issue isn't the label, but knowing which ball is
removed. It may not seem important (since they all look alike), but
consider the following:
1. If a ball is added and removed, then it can't be in the limit set.
2. If a ball is added and not removed, then it must be in the limit
set.
3. No other ball is in the limit set.
If we agree on those three basic principles, then we simply look at it
on a ball-by-ball basis, to see which of those three categories it is
in. (The labels themeselves are not relevant; they are there merely
as convenience for us to talk about them.)
In the original problem, the bin ends up empty because each ball falls
into Category #1: added at t=1/2^[n/10] and removed at t=1/2^n.
In your modified version, only Balls #10n-1 fall into Category #1.
The remainder fall into Category #2 (added at t=1/2^[n/10] but never
removed).
The mistake would be to assume it doesn't matter which ball is
removed. It makes all the difference in the world!
You can modify the problem so that you have 0, a finite number, or an
infinite number of balls left in the bin.
As I pointed out in another post, chosing a ball at random to be
removed, allows you the possibility of any of those, but the measure
of this probability is 1 that it is infinite.
David R Tribble
> Yes, I understand that. It just seems strange that what is left in the
> urn (if anything) should depend on the numbering---although, let me
> repeat---I agree that it does. Why can't we just add and subtract un-
> labeled balls? Of course, in a sense they are still numbered in
> principle, as first, second, third, etc. But what prevents us from
> becoming memory-impaired during the procedure, and thus start removing
> the "wrong" balls by mistake?
Because there is an order involved. The crux of the original
problem is that every ball inserted into the vase is later
removed. This is guaranteed by the numbering of the balls
and the order in which the insertions and removals are
executed.
Without this guarantee, how do you know which ball is
removed and which remains? There is no guarantee that
a given ball is ever removed.
Transfer Principle
> On Jan 22, 2:31 am, Transfer Principle <lwal...@lausd.net> wrote:
> > Which is the main reason I try to look for that equi-consistent
> > set theory -- so that they wouldn't be "cranks" anymore. But
> > unfortunately, finding equi-consistent set theories that
> > satisfying their desiderata is seldom _easy_ -- which is why I
> > haven't been very succesful at doing so _yet_.
> It would certainly be challenging. The main problem is that each ball
> added is removed at some time t=1/2^n. To argue that the limit set is
> non-empty, you have to identify at least one item, and state when it
> was added. I'm not saying there isn't a consistent set theory of some
> kind that would manage it (I'm not a Set Theoretician, so I am not
> educated enough to say), but it would be hard to imagine that one
> would be any less counter-intuitive than the problem you are trying to
> "fix".
But there are so many posters who hold that the vase is full at
time t=0, and apparently, _those_ posters don't consider a full
vase to be more counterintuitive than an empty vase. It's for
the sake of _those_ posters that I seek the alternate theory.
> > Note that it's remotely possible -- depending once again on
> > the axioms -- that one can prove the vase to be nonempty, but
> > one can't actually name (construct) any ball in the vase.
> This is an EXTREMELY good point, one which I have been waiting to hear
> someone mention! It would need to be what is called an omega-
> inconsistent theory (see:
> http://en.wikipedia.org/wiki/%CE%A9-consistent_theory )
Thanks for the information.
> For me though, it is counter-intuitive to have a theory in which a
> property of the natural numbers is individually false for 0, 1,
> 2, ..., n, ..., and false for every natural number, but the statement
> "There exists a natural number with this property" is a theorem.
Agreed.
> With the incredible fragmenting of this thread, I am posting this
> "sequel" for those wishing to continue the discussion.
Do you (Math1723) use Google to access Usenet? If so, then this is
something that always occurs when a thread reaches 1001 posts.
Math1723
>
> But there are so many posters who hold that the vase is full at
> time t=0, and apparently, _those_ posters don't consider a full
> vase to be more counterintuitive than an empty vase. It's for
> the sake of _those_ posters that I seek the alternate theory.
Here is perhaps the area you want to concentrate. By a limit set,
most people assume the following:
1. If a ball is added and removed, then it can't be in the limit set.
2. If a ball is added and not removed, then it must be in the limit
set.
3. No other ball is in the limit set.
If you assume these, then it's pretty straightforward that in the
original problem, the limit set must be empty, since every Ball #n
falls into Category #1 (added at t=1/2^[n/10], removed at t=1/2^n).
On a Ball-by-Ball basis, they are removed.
It seems to me the concept of your limit set must change in this
theory of yours, so that one or more of these assumption fails.
> Do you (Math1723) use Google to access Usenet? If so, then this is
> something that always occurs when a thread reaches 1001 posts.
Yes, that is how I access sci.math. Thanks for your info!
Math1723
>
> Because there is an order involved. The crux of the original
> problem is that every ball inserted into the vase is later
> removed. This is guaranteed by the numbering of the balls
> and the order in which the insertions and removals are
> executed.
>
> Without this guarantee, how do you know which ball is
> removed and which remains? There is no guarantee that
> a given ball is ever removed.
Right. The probability is essentially 0 that you can randomly choose
the correct balls to leave only a finite amount left.
In the end, I think this is the problem a lot of cranks have with this
problem. They confuse ordinal arithmetic with cardinal arithmetic
(because the two are the same in the finite case). They look only at
cardinality and are confused when the answer is different when you
change which balls you take out. Tracking it ball-by-ball, it is
obvious. But sadly, not everyone cares about the logic involved. :-/
David R Tribble
>> It would certainly be challenging. The main problem is that each ball
>> added is removed at some time t=1/2^n. To argue that the limit set is
>> non-empty, you have to identify at least one item, and state when it
>> was added. I'm not saying there isn't a consistent set theory of some
>> kind that would manage it (I'm not a Set Theoretician, so I am not
>> educated enough to say), but it would be hard to imagine that one
>> would be any less counter-intuitive than the problem you are trying to
>> "fix".
>
Transfer Principle wrote:
> But there are so many posters who hold that the vase is full at
> time t=0, and apparently, _those_ posters don't consider a full
> vase to be more counterintuitive than an empty vase. It's for
> the sake of _those_ posters that I seek the alternate theory.
So it did not occur to you that those posters could simply
be wrong?
Do you personally think there is a correct answer to the
problem other than "zero balls remain in the vase"?
Han de Bruijn
The "cranks" are NOT "confused"; they just don't accept your nonsense.
The fact that ordinal arithmetic indeed _is_ identical with cardinal
arithmetic in the finite case, together with the fact that "Infinitum
actu non datur", just means that your so-called logic is VOID of any
meaning. So accusing us of "having a problem" is close to dishonesty,
if not simply fraud.
Han de Bruijn
Dan
All balls have numbers on them . Number gives a ball its identity .
Let's have one of the balls be special , a "multiple identity ball" ,
having on it numbers 1 , 19 , 199 , 1999 , 19999 ......
All other balls have their usual numbers , all different from the
numbers on the multiple identity ball .
The multiple identity ball always stays in the vase , even though , if
they were separate balls , they would all be removed .
Math1723
>
> The "cranks" are NOT "confused"; they just don't accept your nonsense.
It's not "my" nonsense they disagree with. It's the laws of logic.
> The fact that ordinal arithmetic indeed _is_ identical with cardinal
> arithmetic in the finite case, together with the fact that
> "Infinitum actu non datur", just means that your so-called logic is
> VOID of any meaning. So accusing us of "having a problem" is close
> to dishonesty,
> if not simply fraud.
Please identify which step in our reasoning which does not follow from
the axioms and logical rules of inference. (Just as, for example, we
have dome with you.)
Math1723
The labels are not the issue. The issue is: on a ball by ball basis,
does the ball ever get removed from the bin? (The labels are mere
convenience for us to speak about them.) In your example, wouldn't
this Ball get removed at Step #1 since it has a #1 on it (as well as
#19, 199, and all the rest?)
Han de Bruijn
> On Jan 27, 3:40 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > The "cranks" are NOT "confused"; they just don't accept your nonsense.
>
> It's not "my" nonsense they disagree with. It's the laws of logic.
Yes, but your logic is without any meaning. It is VOID.
> > The fact that ordinal arithmetic indeed _is_ identical with cardinal
> > arithmetic in the finite case, together with the fact that
> > "Infinitum actu non datur", just means that your so-called logic is
> > VOID of any meaning. So accusing us of "having a problem" is close
> > to dishonesty, if not simply fraud.
>
> Please identify which step in our reasoning which does not follow from
> the axioms and logical rules of inference. (Just as, for example, we
> have dome with you.)
Your "logic" is no more than a stream of blah blah. And your "axioms"
mean nothing. Why do you think this thread is lasting for so long ?
It's not because you are right and they are wrong. It's because none
of you is telling something sensible. There are no axioms, there are
no rules of inference. Nonsense is all there is. Everybody is making
this up. And there's no way to tell who is right and who is wrong.
At noon. Completed Infinities do not exist. Period.
Han de Bruijn
Math1723
>
> Your "logic" is no more than a stream of blah blah. And your "axioms"
> mean nothing. Why do you think this thread is lasting for so long ?
> It's not because you are right and they are wrong. It's because none
> of you is telling something sensible. There are no axioms, there are
> no rules of inference. Nonsense is all there is. Everybody is making
> this up. And there's no way to tell who is right and who is wrong.
>
> At noon. Completed Infinities do not exist. Period.
If there are no axioms and no rules of inference, no way to tell who
is right and who is wrong, how can you say "completed infinities do
not exist"? By your own argument, you can't know, and so completed
infinities may exist after all, right?
Han de Bruijn
Not right! It is not up to mathematicians to decide whether completed
infinities may exist after all. In order to decide whether they exist
of not, you have to say how they could be observed, eventually, in the
real world, ipse est: in nature. So far, nobody has come up with sort
of experimental setup that would enable such an observation. And thus
it is _reasonable_ to assume that such beasties do not exist. There's
no empirical evidence for gnomes too, huh ..
I hope you realise that completed infinity (a bijection between balls
inserted and balls removed) is the _only_ problem at t=0. Right?
Han de Bruijn
Dan
because
"ball 1" = "ball 19" . So its actual position in the final set would
be indeterminate .
I did not dispute your claims regarding the original problem . I
proposed a variation where one , or some of our balls behave like
"groups of balls" , and see how that affects the problem .
Another way to look at this ball is always being in the vase but
always under one of its different identities (ie until step 19 it has
identity 19 , after which it takes identity 199 (remember that at step
19 , ball 199 is added in the vase , ball 19 is removed ) . After step
199 , it takes on identity 1999 , and so on ) .
Dan
You have an infinitude of vases like so ....
| * * * * * * * ......
In each vase you have a numbered ball , like so :
1 2 3 4 5 6 ....
|* * * * * * ....
At each step you move each ball 1-vase foreword , like so :
1 2 3 4 5 6 ....
|* * * * * * * ....
At the limit step one would have to conclude that all the balls have
disappeared .
Dan
our reference frame from that of the vases , to that of the balls
(ie , the balls are not going foreword , the vases are going
backward ) .
Math1723
> Here are two other related problems :
> You have an infinitude of vases like so ....
>
> | * * * * * * * ......
> In each vase you have a numbered ball , like so :
For convenience, let's label each Vase as Vase #1, Vase #2, etc.
(Obviously the label is not important, but it is a convenience so that
I may refer to them individually.)
To match the assumption of the original problem, I will assume that
these are the only vases, that there is no Vase with a Transfinite
label, is that right?
> 1 2 3 4 5 6 ....
> |* * * * * * ....
>
> At each step you move each ball 1-vase foreword , like so :
>
> 1 2 3 4 5 6 ....
> |* * * * * * * ....
In other words, at each Step #k, Ball #n is moved from Vase #n+k-1 to
Vase #n+k, for all n. This is important, as there are two steps here:
1) Ball #n is removed from Vase #n+k-1, and Ball #n is placed into
Vase #n+k.
> At the limit step one would have to conclude that all the balls have
> disappeared .
Or better stated, no Ball #n is in any Vase #k. Now, the problem does
not state what to do at t=0 at when there are no more vases to put the
balls in. Presumably, they stay removed (as opposed to, say, put them
all back in their original vases, create a new Vase to put them all
in, or any number of other possibilities). Saying they "disappeared"
is stating more than I can tell from the problem you described. If
you are saying that any ball with no more vases to go into must be
destroyed, then perhaps so.
Math1723
>
> I hope you realise that completed infinity (a bijection between balls
> inserted and balls removed) is the _only_ problem at t=0. Right?
I am not seeing that as the problem. I am seeing as the problem the
basic failure of accepting the rules of logic: given a set of
premises, certain conclusions must follow. You seem to be hung up on
the truth or falsity of the premises (mathematical axioms), which is
not pertinent. What is pertinent is that the conclusion necessarily
follows the assumptions.
Starting from false premises is a common Mathematical tool, used in
Proof by Contradiction. You may even begin with with premises which
you do not necessarily believe, just to see what comes out from it.
For example, I personally have no problem "believing" in the Axiom of
Choice, but that does not prevent me from studying and enjoying the
results from assuming its falsehood. (After all, that process is how
we arrived at Non-Euclidean Geometry.)
Upon agreement with that, then it then merely boils down to: which
assumptions (axioms) you accept to arrive at your results (theorems).
Perhaps you have a problem with ZFC's Axiom of Infinity? Or perhaps
the Axiom Schema of Specification? (Se:
http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory for
more details.) That's fine, you are free to believe or not believe
whatever you like. (Mathematicians, as a rule, don't require your
position on the Deity of Christ, either.) What they do care about are
your starting assumptions and rules of inference.
Dan
I think we agree here . There is no sufficient reason at limit for
them to be at any one position in space as opposed to the other ,
based solely on the information of the previous steps . We may even
say that at limit it is possible for them to appear exactly at their
initial position ie
1 2 3 4 5 6 ....
|* * * * * * ....
unless we specify that this is not the case , which is my point (any
number of other possibilities) .
Marshall
On Jan 27, 12:40 am, Han de Bruijn <umum...@gmail.com> wrote:
> The "cranks" are NOT "confused" [...]
but then you say:
> The fact that ordinal arithmetic indeed _is_ identical with cardinal
> arithmetic in the finite case, together with the fact that "Infinitum
> actu non datur", just means that your so-called logic is VOID of any
> meaning.
which shows you are indeed a confused crank.
Marshall
Marshall
>
> Your "logic" is no more than a stream of blah blah. And your "axioms"
> mean nothing.
Jan 27, 2011: Han de Bruijn officially rejects logic.
Marshall
William Hughes
> I hope you realise that completed infinity (a bijection between balls
> inserted and balls removed) is the _only_ problem at t=0. Right?
>
Nope.
There is a strange belief among many who do not like the paradoxes
of infinity the if they intone the word "completed" the paradoxes
will go away. However, the paradoxes can be described in terms
of "potential infinity" so they do not go away.
In the present case. (Note that a function on a potentially infinite
set is a potentially infinite set of ordered pairs.)
Define vase(t) mapping the rational numbers into finite sets of
natural numbers
by
vase(t) = {} if t < -1 or if t >=0
{1,2,3,...,10*n}\{1,2,3,...,n-1} (0-1/n) <= t <=
(0-1/(n+1)) for any natural n
Define num_vase(t) mapping the rational numbers into the natural
numbers U {0}
num_vase(t) = number of elements in the set vase(t)
Define a potentially infinite set of functions state_n(t) from the
rationals to {0,1}
state_n(t) = 1 if (1-1/n) <= t <= (1-1/(n+1)), 0 otherwise.
No completed infinite sets anywhere.
It is easy to check that if n is a natural number
n is an element of vase(t) iff state_n(t)=1.
It is easy to check that if n is a natural number
state_n is continuous at t=0
It is easy to see that num_vase(t) increases without bound as
we approach 0 from below.
It is easy to see that the first t>-1 for which num_vase(t) = 0
is 0.
Informally, num_vase drops to 0 at t=0 even though nothing
happens at t=0.
If you want to get rid of the paradoxes you must get
rid of potential infinity as well. This means that you
have to put an upper bound on the naturals.
- William Hughes
David R Tribble
> The fact that ordinal arithmetic indeed _is_ identical with cardinal
> arithmetic in the finite case, together with the fact that
> "Infinitum actu non datur", just means that your so-called logic is
> VOID of any meaning. So accusing us of "having a problem" is close
> to dishonesty, if not simply fraud.
>
> mean nothing. Why do you think this thread is lasting for so long ?
> It's not because you are right and they are wrong. It's because none
> of you is telling something sensible. There are no axioms, there are
> no rules of inference. Nonsense is all there is. Everybody is making
> this up. And there's no way to tell who is right and who is wrong.
>
> At noon. Completed Infinities do not exist. Period.
"God created infinity, and man, unable to understand infinity,
created finite sets."
(From an interesting collection of mathetical quotes and cartoons
at: http://math.sfsu.edu/beck/quotes.html )
David R Tribble
>> Completed Infinities do not exist. Period.
>
Math1723 wrote:
>> If there are no axioms and no rules of inference, no way to tell who
>> is right and who is wrong, how can you say "completed infinities do
>> not exist"? By your own argument, you can't know, and so completed
>> infinities may exist after all, right?
>
Han de Bruijn wrote:
> Not right! It is not up to mathematicians to decide whether completed
> infinities may exist after all. In order to decide whether they exist
> of not, you have to say how they could be observed, eventually, in the
> real world, ipse est: in nature.
I guess that explains why quaternions exist.
Virgil
Virgil
<3b2acf3e-802d-4e77...@x4g2000yql.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
> On Jan 27, 2:21 pm, Math1723 <anonym1...@aol.com> wrote:
> > On Jan 27, 8:10 am, Han de Bruijn <umum...@gmail.com> wrote:
> >
> > > Your "logic" is no more than a stream of blah blah. And your "axioms"
> > > mean nothing. Why do you think this thread is lasting for so long ?
> > > It's not because you are right and they are wrong. It's because none
> > > of you is telling something sensible. There are no axioms, there are
> > > no rules of inference. Nonsense is all there is. Everybody is making
> > > this up. And there's no way to tell who is right and who is wrong.
> >
> > > At noon. Completed Infinities do not exist. Period.
> >
> > If there are no axioms and no rules of inference, no way to tell who
> > is right and who is wrong, how can you say "completed infinities do
> > not exist"? By your own argument, you can't know, and so completed
> > infinities may exist after all, right?
>
> Not right! It is not up to mathematicians to decide whether completed
> infinities may exist after all. In order to decide whether they exist
> of not, you have to say how they could be observed, eventually, in the
> real world, ipse est: in nature.
That is physics, not mathematics. Requiring physical evidence is no part
of mathematics.
Those physicists who demand that mathematicians be their slaves are
losers.
MoeBlee
> It is not up to mathematicians to decide whether completed
> infinities may exist after all. In order to decide whether they exist
> of not, you have to say how they could be observed, eventually, in the
> real world, ipse est: in nature.
That's your DOGMA.
Moreover, merely working in set theory does not carry a commitment
that anything exists in any sense other than - as we do VERIFY by
finite inspection of finite strings (proofs) of finite finite strings
(formulas) - a certain formula with an existential quantifier at front
is of a certain kind (theorem).
> So far, nobody has come up with sort
> of experimental setup that would enable such an observation.
I don't know anyone who has observed the number 0, the real number pi,
nor the Euclidean plane.
MoeBlee
MoeBlee
>
> The fact that ordinal arithmetic indeed _is_ identical with cardinal
> arithmetic in the finite case, together with the fact that "Infinitum
> actu non datur", just means that your so-called logic is VOID of any
> meaning.
Logic is used for making inferences or proving statements. Meaning is
a property of statements themselves.
Do you think that your mathematical arguments adhere to logic? If so,
would you point to some codification of your logic so that we can
objectively check that your arguments do adhere to that logic?
MoeBlee
Han de Bruijn
Quaterions are quadruples of _reals_ with certain rules for enabling
the "elementary operations" on them. What's your problem?
Han de Bruijn
Han de Bruijn
Everybody has observed _approximations_ of the number 0, the real
number pi and the Euclidean plane. And yes, everybody has observed
approximations of _infinity_ as well, which is the _limit_ concept,
as has been developed by Gauss and Cauchy. Quite fortunately, the
approximations of the number 0, the real number pi and the Euclidean
plane are in agreement with the abstract stuff, within error bounds.
But quite UN-fortunately, completed infinity is NOT in agreement with
the proper approaches to infinity via limits.
Han de Bruijn
Ilmari Karonen
> On Jan 27, 9:08 am, Dan <dan.ms.ch...@gmail.com> wrote:
>> Here are two other related problems :
>> You have an infinitude of vases like so ....
>>
>> | * * * * * * * ......
>> In each vase you have a numbered ball , like so :
>
> For convenience, let's label each Vase as Vase #1, Vase #2, etc.
> (Obviously the label is not important, but it is a convenience so that
> I may refer to them individually.)
>
> To match the assumption of the original problem, I will assume that
> these are the only vases, that there is no Vase with a Transfinite
> label, is that right?
>
>> 1 2 3 4 5 6 ....
>> |* * * * * * ....
>>
>> At each step you move each ball 1-vase foreword , like so :
>>
>> 1 2 3 4 5 6 ....
>> |* * * * * * * ....
>
> In other words, at each Step #k, Ball #n is moved from Vase #n+k-1 to
> Vase #n+k, for all n. This is important, as there are two steps here:
> 1) Ball #n is removed from Vase #n+k-1, and Ball #n is placed into
> Vase #n+k.
Actually, it does not particularly matter whether you decompose the
movement steps like that or not. You can even add a non-zero time
interval between taking the balls out of the vases and putting them
back in, as long as you make it short enough that the balls are back
in the vases before they get taken out again the next time. The
outcome (and the paradox inherent in it) is still the same.
>> At the limit step one would have to conclude that all the balls have
>> disappeared .
>
> Or better stated, no Ball #n is in any Vase #k. Now, the problem does
> not state what to do at t=0 at when there are no more vases to put the
> balls in. Presumably, they stay removed (as opposed to, say, put them
> all back in their original vases, create a new Vase to put them all
> in, or any number of other possibilities).
Didn't you just fall into your own trap? The problem doesn't say what
is done at t=0, so, implictly, nothing is done at t=0. Indeed there's
not much to do, since at t=0 there are no balls in any of the vases to
take them out of.
A better question is, of course, where the balls are at t=0 if all the
vases are empty. The answer is the same as in the single-ball variant
discussed in one branch of the earlier thread: the location of the
balls at t=0 is not well defined.
Of course you could modify the problem to add an extra step at t=0 in
which the balls are placed anywhere you want them to be, but then the
big question becomes where they were _before_ you moved them at t=0.
> Saying they "disappeared" is stating more than I can tell from the
> problem you described. If you are saying that any ball with no more
> vases to go into must be destroyed, then perhaps so.
That is correct, but stating that they are outside the vases at t=0 is
also not implied by the problem as stated, and indeed contradicts it:
depending on how you assume the moves are carried out, either the
balls are never taken out of the vases, or every time they are taken
out they're also subsequently put back in, all before t=0.
--
Ilmari Karonen
To reply by e-mail, please replace ".invalid" with ".net" in address.
LudovicoVan
> On Jan 27, 2:40 am, Han de Bruijn <umu...@gmail.com> wrote:
> >
> > The fact that ordinal arithmetic indeed _is_ identical with cardinal
> > arithmetic in the finite case, together with the fact that "Infinitum
> > actu non datur", just means that your so-called logic is VOID of any
> > meaning.
>
> Logic is used for making inferences or proving statements. Meaning is
> a property of statements themselves.
Logic is not mathematical logic.
-LV
Marshall
This is just bullshit. Sorry for you.
Marshall
Ilmari Karonen
> On Jan 25, 7:35 pm, David R Tribble <da...@tribble.com> wrote:
>> Ray Vickson wrote:
>> > In a previous post about this I posited the following scenario (based
>> > on a Wiki article): to drive yourself crazy, look at the modification
>> > in which at t = 1 we put into the urn balls 1--9; at t = 1/2 sec we
>> > add balls 10--19 and remove ball 9; at t = 1/4 we add balls 20--29 and
>> > remove ball 19, etc. NOW there are plenty of balls at t = 0. Aside
>> > from re-numbering, the exact same add/remove operations are performed
>> > in each case!
>>
>> Well, exactly the same _number_ of add/remove operations
>> are performed, but a _different_ set of balls are removed than
>> in the original problem. And a completely different set of balls
>> are never removed from the vase.
>>
>> So yours is a completely different problem.
>
> Yes, I understand that. It just seems strange that what is left in the
> urn (if anything) should depend on the numbering---although, let me
> repeat---I agree that it does. Why can't we just add and subtract un-
> labeled balls? Of course, in a sense they are still numbered in
> principle, as first, second, third, etc. But what prevents us from
> becoming memory-impaired during the procedure, and thus start removing
> the "wrong" balls by mistake?
Note that it doesn't actually make any difference if we swap the
labels on the balls currently in the vase at any given time. The
outcome only changes if we either a) swap labels between between balls
currently in the vase and those currently outside it, or b) carry out
an infinite number of swaps within the vase.
Indeed, that's sort of obvious -- the only ways to keep a ball from
being removed are to either swap its label with a ball already removed
or to keep swapping its label with a ball not yet removed whenever
it's about to be removed.
FredJeffries
>
> "God created infinity, and man, unable to understand infinity,
> created finite sets."
>
> (From an interesting collection of mathetical quotes and cartoons
> at:http://math.sfsu.edu/beck/quotes.html)
I like it.
Han de Bruijn
> On Jan 27, 9:48 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > I hope you realise that completed infinity (a bijection between balls
> > inserted and balls removed) is the _only_ problem at t=0. Right?
>
> Nope.
>
> There is a strange belief among many who do not like the paradoxes
> of infinity the if they intone the word "completed" the paradoxes
> will go away. However, the paradoxes can be described in terms
> of "potential infinity" so they do not go away.
>
> In the present case. (Note that a function on a potentially infinite
> set is a potentially infinite set of ordered pairs.)
>
> Define vase(t) mapping the rational numbers into finite sets of
> natural numbers
> by
>
> vase(t) = {} if t < -1 or if t >=0
> {1,2,3,...,10*n}\{1,2,3,...,n-1} (0-1/n) <= t <=
> (0-1/(n+1)) for any natural n
So you are right _by definition_: vase(t >= 0) = {} . Ookkaayy ..
> Define num_vase(t) mapping the rational numbers into the natural
> numbers U {0}
>
> num_vase(t) = number of elements in the set vase(t)
>
> Define a potentially infinite set of functions state_n(t) from the
> rationals to {0,1}
>
> state_n(t) = 1 if (1-1/n) <= t <= (1-1/(n+1)), 0 otherwise.
>
> No completed infinite sets anywhere.
>
> It is easy to check that if n is a natural number
> n is an element of vase(t) iff state_n(t)=1.
>
> It is easy to check that if n is a natural number
> state_n is continuous at t=0
Sure. If a function junps from 1 to 0 then it is continuous ..
> It is easy to see that num_vase(t) increases without bound as
> we approach 0 from below.
>
> It is easy to see that the first t>-1 for which num_vase(t) = 0
> is 0.
Yeah, because you are right by definition. Easy huh?
> Informally, num_vase drops to 0 at t=0 even though nothing
> happens at t=0.
> If you want to get rid of the paradoxes you must get
> rid of potential infinity as well. This means that you
> have to put an upper bound on the naturals.
Let's just do that. Let {0,1,2,3,4 .. n} be an initial segment of the
naturals. There is an upper bound, namely (n). Now take the limit for
n -> oo in some context, where such is useful. Do you still have that
upper bound then?
Han de Bruijn
Han de Bruijn
Ok. This argument of yours is so convincing that I have to capitulate.
Han de Bruijn
Han de Bruijn
> On Jan 27, 8:48 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > I hope you realise that completed infinity (a bijection between balls
> > inserted and balls removed) is the _only_ problem at t=0. Right?
>
> I am not seeing that as the problem. I am seeing as the problem the
> basic failure of accepting the rules of logic: given a set of
> premises, certain conclusions must follow. You seem to be hung up on
> the truth or falsity of the premises (mathematical axioms), which is
> not pertinent. What is pertinent is that the conclusion necessarily
> follows the assumptions.
Denied. The bijection between balls inserted and balls removed, thus
completed infinity, is the only problem. Do you agree that there IS
such a bijection? Please save me the rest of your "logic". Yes or No
is quite a sufficient answer.
> Starting from false premises is a common Mathematical tool, used in
> Proof by Contradiction. You may even begin with with premises which
> you do not necessarily believe, just to see what comes out from it.
> For example, I personally have no problem "believing" in the Axiom of
> Choice, but that does not prevent me from studying and enjoying the
> results from assuming its falsehood. (After all, that process is how
> we arrived at Non-Euclidean Geometry.)
>
> Upon agreement with that, then it then merely boils down to: which
> assumptions (axioms) you accept to arrive at your results (theorems).
> Perhaps you have a problem with ZFC's Axiom of Infinity? Or perhaps
> the Axiom Schema of Specification? (Se:http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryfor
> more details.) That's fine, you are free to believe or not believe
> whatever you like. (Mathematicians, as a rule, don't require your
> position on the Deity of Christ, either.) What they do care about are
> your starting assumptions and rules of inference.
Yes, yes, I have no bananas.
Han de Bruijn
Han de Bruijn
> In article
> <b2db85d3-b829-4bdf-9e52-a88c1093d...@k22g2000yqh.googlegroups.com>,
> Han de Bruijn <umum...@gmail.com> wrote:
>
>
>
>
>
> > On Jan 27, 1:43 pm, Math1723 <anonym1...@aol.com> wrote:
> > > On Jan 27, 3:40 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > > > The "cranks" are NOT "confused"; they just don't accept your nonsense.
>
> > > It's not "my" nonsense they disagree with. It's the laws of logic.
>
> > Yes, but your logic is without any meaning. It is VOID.
>
> > > > The fact that ordinal arithmetic indeed _is_ identical with cardinal
> > > > arithmetic in the finite case, together with the fact that
> > > > "Infinitum actu non datur", just means that your so-called logic is
> > > > VOID of any meaning. So accusing us of "having a problem" is close
> > > > to dishonesty, if not simply fraud.
>
> > > Please identify which step in our reasoning which does not follow from
> > > the axioms and logical rules of inference. (Just as, for example, we
> > > have dome with you.)
>
> > Your "logic" is no more than a stream of blah blah. And your "axioms"
> > mean nothing. Why do you think this thread is lasting for so long ?
> > It's not because you are right and they are wrong. It's because none
> > of you is telling something sensible. There are no axioms, there are
> > no rules of inference. Nonsense is all there is. Everybody is making
> > this up. And there's no way to tell who is right and who is wrong.
>
> > At noon. Completed Infinities do not exist. Period.
>
Sometimes I wish you were right ..
Han de Bruijn
William Hughes
> On Jan 27, 7:29 pm, William Hughes <wpihug...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On Jan 27, 9:48 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > > I hope you realise that completed infinity (a bijection between balls
> > > inserted and balls removed) is the _only_ problem at t=0. Right?
>
> > Nope.
>
> > There is a strange belief among many who do not like the paradoxes
> > of infinity the if they intone the word "completed" the paradoxes
> > will go away. However, the paradoxes can be described in terms
> > of "potential infinity" so they do not go away.
>
> > In the present case. (Note that a function on a potentially infinite
> > set is a potentially infinite set of ordered pairs.)
>
> > Define vase(t) mapping the rational numbers into finite sets of
> > natural numbers
> > by
>
> > vase(t) = {} if t < -1 or if t >=0
> > {1,2,3,...,10*n}\{1,2,3,...,n-1} (0-1/n) <= t <=
> > (0-1/(n+1)) for any natural n
>
> So you are right _by definition_: vase(t >= 0) = {} . Ookkaayy ..
>
>
A definition is neither right nor wrong. What is important
is that this definition does not use completed infinity.
>
>
>
>
>
>
>
> > Define num_vase(t) mapping the rational numbers into the natural
> > numbers U {0}
>
> > num_vase(t) = number of elements in the set vase(t)
>
> > Define a potentially infinite set of functions state_n(t) from the
> > rationals to {0,1}
>
> > state_n(t) = 1 if (1-1/n) <= t <= (1-1/(n+1)), 0 otherwise.
>
> > No completed infinite sets anywhere.
>
> > It is easy to check that if n is a natural number
> > n is an element of vase(t) iff state_n(t)=1.
>
> > It is easy to check that if n is a natural number
> > state_n is continuous at t=0
>
> Sure. If a function junps from 1 to 0 then it is continuous ..
I think you are confusing state_n and num_vase
>
> > It is easy to see that num_vase(t) increases without bound as
> > we approach 0 from below.
>
> > It is easy to see that the first t>-1 for which num_vase(t) = 0
> > is 0.
>
> Yeah, because you are right by definition. Easy huh?
The definitions I give do not use
completed infinity but still lead to the same
paradox.
num_vase drops to 0 at t=0 even though,
there is no number n
for which state_n changes at t=0
- William Hughes
Han de Bruijn
How objectively is a thread that exceeds 1001 postings without a shred
of agreement between its debaters? I've setup a thread about something
_much_ more complicated than this. And got not a single objection:
https://groups.google.com/group/sci.math/msg/b06347def941d685?hl=en
Probably because my "mathematical arguments adhere to logic" indeed?
Han de Bruijn
Han de Bruijn
<shrug>
Han de Bruijn
William Hughes
Perhaps you would prefer the phrasing
num_vase drops to 0 at t=0 even though,
there is no number n
for which n enters or leaves the
set vase(t) at t=0.
No completed infinity, same paradox.
This certainly contradicts your statement
"completed infinity
(a bijection between balls
inserted and balls removed) is
the _only_ problem at t=0"
- William Hughes
Math1723
>
> Denied. The bijection between balls inserted and balls removed, thus
> completed infinity, is the only problem. Do you agree that there IS
> such a bijection? Please save me the rest of your "logic". Yes or No
> is quite a sufficient answer.
Since the set of balls inserted is the same as the set of balls
removed, and any set will have a bijection onto itself, then the
answer is: Yes.
Virgil
<52592723-6469-4afb...@x4g2000yql.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
That wish has been granted. Nobody's world is The World.
Virgil
<1b952e6c-e6e5-4350...@e21g2000yqe.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
> On Jan 27, 3:55�pm, Math1723 <anonym1...@aol.com> wrote:
> > On Jan 27, 8:48�am, Han de Bruijn <umum...@gmail.com> wrote:
> >
> > > I hope you realise that completed infinity (a bijection between balls
> > > inserted and balls removed) is the _only_ problem at t=0. Right?
> >
> > I am not seeing that as the problem. �I am seeing as the problem the
> > basic failure of accepting the rules of logic: given a set of
> > premises, certain conclusions must follow. �You seem to be hung up on
> > the truth or falsity of the premises (mathematical axioms), which is
> > not pertinent. �What is pertinent is that the conclusion necessarily
> > follows the assumptions.
>
> Denied. The bijection between balls inserted and balls removed, thus
> completed infinity, is the only problem. Do you agree that there IS
> such a bijection? Please save me the rest of your "logic". Yes or No
> is quite a sufficient answer.
According to any standard definition of bijection, yes!
>
> > Starting from false premises is a common Mathematical tool, used in
> > Proof by Contradiction. �You may even begin with with premises which
> > you do not necessarily believe, just to see what comes out from it.
> > For example, I personally have no problem "believing" in the Axiom of
> > Choice, but that does not prevent me from studying and enjoying the
> > results from assuming its falsehood. �(After all, that process is how
> > we arrived at Non-Euclidean Geometry.)
> >
> > Upon agreement with that, then it then merely boils down to: which
> > assumptions (axioms) you accept to arrive at your results (theorems).
> > Perhaps you have a problem with ZFC's Axiom of Infinity? �Or perhaps
> > the Axiom Schema of Specification?
> > �(Se:http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryfor
> > more details.) �That's fine, you are free to believe or not believe
> > whatever you like. �(Mathematicians, as a rule, don't require your
> > position on the Deity of Christ, either.) �What they do care about are
> > your starting assumptions and rules of inference.
>
> Yes, yes, I have no bananas.
>
> Han de Bruijn
You merely are bananas.
Virgil
<8920d2aa-b428-477b...@s11g2000yqh.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
They do not exist in any "real" world but only in mathematically
imagined worlds. So they cannot exist in your world, at least as you
have often defined it.
Marshall
Obviously merely calling out someone's bullshit is not an argument per
se.
So, this approximation of pi you've observed: how much did it
weigh? How did it smell? If it didn't have any such characteristics,
then it wasn't physical, so your claim of observing it fails. If
it did have these characteristics, then it wasn't an abstract
quantity, and so your claim also fails.
Like many cranks, you fail to understand abstraction. Perhaps
this is a heretofore unidentified cognitive defect.
Marshall
Marshall
You also have no axioms nor rules of inference. You are
well-stocked with bravado however!
Marshall
Marshall
> On Jan 28, 2:49 am, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Jan 27, 2:40 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > > The fact that ordinal arithmetic indeed _is_ identical with cardinal
> > > arithmetic in the finite case, together with the fact that "Infinitum
> > > actu non datur", just means that your so-called logic is VOID of any
> > > meaning.
>
> > Logic is used for making inferences or proving statements. Meaning is
> > a property of statements themselves.
>
> > Do you think that your mathematical arguments adhere to logic? If so,
> > would you point to some codification of your logic so that we can
> > objectively check that your arguments do adhere to that logic?
>
> How objectively is a thread that exceeds 1001 postings without a shred
> of agreement between its debaters? I've setup a thread about something
> _much_ more complicated than this. And got not a single objection:
Shall we assume from this dodge of Moe's question that you admit
that your arguments here do not adhere to logic? Or would you
care to answer it?
Marshall
MoeBlee
>
> Everybody has observed _approximations_ of the number 0, the real
> number pi and the Euclidean plane.
I've never observed such a critter with my eyes, ears, nose, tongue,
or skin.
Can you tell me where one of those approxiomations is? Are there any
in my kitchen, backyard, or maybe at the dry cleaners down the street?
Do they move around? How fast do they travel when they do?
MoeBlee
MoeBlee
I didn't say it is.
MoeBlee
MoeBlee
> On Jan 28, 2:49 am, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Jan 27, 2:40 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > > The fact that ordinal arithmetic indeed _is_ identical with cardinal
> > > arithmetic in the finite case, together with the fact that "Infinitum
> > > actu non datur", just means that your so-called logic is VOID of any
> > > meaning.
>
> > Logic is used for making inferences or proving statements. Meaning is
> > a property of statements themselves.
>
> > Do you think that your mathematical arguments adhere to logic? If so,
> > would you point to some codification of your logic so that we can
> > objectively check that your arguments do adhere to that logic?
> Probably because my "mathematical arguments adhere to logic" indeed?
I asked if you would show a codification of your logic. 'Because my
mathematical arguments adhere to logic"' is not an answer to my
question.
MoeBlee
Han de Bruijn
Thanks. Then the disagreement between me and you is insurmountable.
No hard feelings on my side. It's exactly what has to be expected.
Later.
Han de Bruijn
Han de Bruijn
Nobody is perfect. Am I nobody?
Han de Bruijn
Han de Bruijn
What if I calculate three dimensional rotations with them?
Han de Bruijn
Han de Bruijn
: simple Simon says .. Weight and smell ARE abstractions, dummy ..
> Like many cranks, you fail to understand abstraction. Perhaps
> this is a heretofore unidentified cognitive defect.
There is a difference between Abstraction and Idealization. For you,
such a distinction simply does not exist. That makes your world a bit
less complicated than mine.
Han de Bruijn
Marshall
Han de Bruijn
Yes! They are at any place you've mentioned. Once you have developed
an eye for them, you will see them. Mathematics really IS everywhere!
Han de Bruijn
Han de Bruijn
No need to anwer that question. It has been settled in my debate with
Math1723 that taking for granted the bijection between balls inserted
and balls removed at t=0 is the _only_ thing I have to object against.
For the rest, _all_ my "mathematical arguments adhere to logic".
Han de Bruijn
Han de Bruijn
Then I'm perfect! Great relief ..
Han de Bruijn
William Hughes
>
> It has been settled in my debate with
> Math1723 that taking for granted the bijection between balls inserted
> and balls removed at t=0 is the _only_ thing I have to object against.
Since this bijection can be expressed in terms of potential infinity,
you must either accept the bijection or reject potential infinity.
To reject potential infinity is to put a bound on the natural numbers.
- William Hughes
Marshall
So, when I merely call you on your bullshit, you complain there
is no argument. When I present an argument, you dodge it.
Your rhetorical skills are good; your logical ones not so much.
In any event, the claim that you've observed mathematical
abstractions is plainly false.
Marshall
MoeBlee
So I can go to my dry cleaner and the number pi will be there for me
to see. And it must be a really big thing if it's at my dry cleaner
and also in your coat pocket at the same time. Funny that something
that big I've never seen. Maybe I need to see an eye doctor. Or, maybe
you need to see a completely different kind of doctor.
MoeBlee
MoeBlee
Well, unless you tell me more about your "logic" than I don't know
what you mean when you refer to it. So, at least for me, there is a
need for an answer to my question. Anyway, as I mentioned, the word
"because" in your previous reply just didn't make sense the way you
used it.
MoeBlee
MoeBlee
And you're Jack Lemmon in "Some Like It Hot" too. Nicely done!
MoeBlee
Virgil
<92dc71f5-5ecc-41c9...@x3g2000yqj.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
Mathematics in this world is, like beauty, only in the eye of the
beholder, not out there among the beheld.
Virgil
<48c03e4d-f58a-4306...@d28g2000yqc.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
Such calculations are mental constructs, at least in principle, even if
analogues of them can be carried out physically, thus no part of any
physical world.
Just as all mathematics is inherently mental even when applied to the
physical world. The application is translation of the mental onto the
physical, and is not any essential part of the mathematics itself.
Virgil
<22a0516f-30df-4b08...@b34g2000yqc.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
Possibly a nobody, but not that Nobody who is perfect.
>
> Han de Bruijn
Virgil
<c41ab4e9-324d-42c8...@z3g2000yqk.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
While Nobody many be perfect, nobody is not.
Virgil
<667efe5e-0b77-4fdb...@r16g2000yql.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
It does not HAVE to be expected, but was to be.
David R Tribble
>> Not right! It is not up to mathematicians to decide whether completed
>> infinities may exist after all. In order to decide whether they exist
>> of not, you have to say how they could be observed, eventually, in the
>> real world, ipse est: in nature.
>
David R Tribble wrote:
>> I guess that explains why quaternions exist.
>
Han de Bruijn wrote:
> Quaterions are quadruples of _reals_ with certain rules for enabling
> the "elementary operations" on them. What's your problem?
Yes, so according to you, they don't exist.
David R Tribble
>> Do you think that your mathematical arguments adhere to logic? If so,
>> would you point to some codification of your logic so that we can
>> objectively check that your arguments do adhere to that logic?
>
Han de Bruijn wrote:
> How objectively is a thread that exceeds 1001 postings without a shred
> of agreement between its debaters?
Which debaters are those?
I count agreement among all but three of the posters,
and one more voting "present".
> I've setup [sic] a thread about something
> _much_ more complicated than this. And got not a single objection:
Therefore proving it is correct.
David R Tribble
>> It is not up to mathematicians to decide whether completed
>> infinities may exist after all. In order to decide whether they exist
>> of not, you have to say how they could be observed, eventually, in the
>> real world, ipse est: in nature.
>
David R Tribble wrote:
>> I guess that explains why quaternions exist.
>
Han de Bruijn wrote:
>> Quaterions are quadruples of _reals_ with certain rules for enabling
>> the "elementary operations" on them. What's your problem?
>
Virgil wrote:
> They do not exist in any "real" world but only in mathematically
> imagined worlds. So they cannot exist in your world, at least as you
> have often defined it.
Han de Bruijn wrote:
> What if I calculate three dimensional rotations with them?
What if you do? How does that make them exist as
physical entities in the real world? Would that magically
convert the real values of quaternions into something other
than approximations?
Han de Bruijn
Potential Infinity is not a defined concept in standard mathematics.
(And I am not going to define it either) So your argument is _void_:
this bijection can NOT be expressed in terms of potential infinity.
Han de Bruijn
Han de Bruijn
No, you need to develop more skills in making obvious abstractions.
I had poison in my bladder and LOGARITHMS have helped me to prevent
suffering from side effects; that's how I could _avoid_ visiting a
doctor. Mathematics has enabled me to setup a recipe for a dealing
with this disease (cancer), and to survive it:
http://hdebruijn.soo.dto.tudelft.nl/jaar2009/spoeling.pdf
Han de Bruijn
Han de Bruijn
Aren't approximations all we can possibly hope for?
Han de Bruijn
William Hughes
You claimed that the *only* problem was due to completed infinity.
I noted that the bijection can be described
using arbitrarily large finite sets.
The problem is with the concept of an
unbounded entity, not with the
concept of "completed" or "potential"
or whatever nomenclature you
want to use.
You want to have your cake and eat it.
- William Hughes
Han de Bruijn
This rather is a soup of words, so drink it, I'd say.
But maybe you're quite right about the concept of an unbounded entity.
It has sunk into my mind, which means that you actually have a point.
Maybe I'm going to think about it too.
Later.
Han de Bruijn
Marshall
>
> I had poison in my bladder and LOGARITHMS have helped me to prevent
> suffering from side effects; that's how I could _avoid_ visiting a
> doctor. Mathematics has enabled me to setup a recipe for a dealing
> with this disease (cancer), and to survive it:
Excuse me, but did you just say that you had cancer and
you *didn't* go see a doctor?
Marshall
MoeBlee
> On Jan 29, 9:39 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> > On Jan 29, 9:56 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > > On Jan 28, 10:57 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > > > On Jan 28, 4:27 am, Han de Bruijn <umum...@gmail.com> wrote:
>
> > > > > Everybody has observed _approximations_ of the number 0, the real
> > > > > number pi and the Euclidean plane.
>
> > > > I've never observed such a critter with my eyes, ears, nose, tongue,
> > > > or skin.
>
> > > > Can you tell me where one of those approxiomations is? Are there any
> > > > in my kitchen, backyard, or maybe at the dry cleaners down the street?
> > > > Do they move around? How fast do they travel when they do?
>
> > > Yes! They are at any place you've mentioned. Once you have developed
> > > an eye for them, you will see them. Mathematics really IS everywhere!
>
> > So I can go to my dry cleaner and the number pi will be there for me
> > to see. And it must be a really big thing if it's at my dry cleaner
> > and also in your coat pocket at the same time. Funny that something
> > that big I've never seen. Maybe I need to see an eye doctor. Or, maybe
> > you need to see a completely different kind of doctor.
>
> No, you need to develop more skills in making obvious abstractions.
Abstractions are not at issue. You were arguing that such things are
PHYSICALLY observable (e.g. by sight). I do pretty well conceiving
abstract objects such as 0, pi, and the Euclidean plane. But that
doesn't make them nor approximations of them objects that I see with
my eyes.
Once again we come full circle in the revolving door of Han de Bruijn.
MoeBlee
Han de Bruijn
Not for the _adverse effects_ other people usually have, caused by the
treatment (chemotherapy) the doctors give us. After I've found out how
to prevent them (thanks to mathematics!), I didn't suffer anymore from
any adverse effects. And could go back to work, after one day of being
a little bit drunk (: due to the diuretics = beer).
Han de Bruijn
Han de Bruijn
So you're not only deaf for sane arguments, but you're blind as well.
Han de Bruijn