Urns of Infinity

[pikachamp]

page infinity
i mean five


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[This message has been edited by pikachamp (edited 05-22-2000).]

[CB]

OK, page FIVE is enough! STOP the discussions!!! I HAVE INSIDE INFO on this one and it's D#2 that's been lazy. Just STOP the arguing or go to a dedicated topic...

[Wonko the Sane]

Hmm...an insider with unregistered status... nevermind. Although the point is valid, we'll see the solution soon and all possible solutions seem to have been posted, so I say we just wait and bounce in our seats.

[worm]

if i were a betting man, i'd say that insider was crystyb, but i'm not a betting man so i won't say that

[worm]

if i were a betting man, i'd say that insider was crystyb, but i'm not a betting man so i won't say that

[worm]

now, i know how murray feels

[ZenBeam]

The Minotaur says he fixed the demon numbering problem, but I've reloaded the question, and it looks the same to me. Can anyone see what he changed, or did he forget to upload the update?

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It is so clear, and so it is hard to see.

[new_on_site]

Hello everyone,

have read your 5 pages of comments with great interest. Let me give it a try:

The only question is: did demon 1 cheat?

I say *yes* - because

according to the text he

1)
put balls 1-9 into the urn
2)
made a 10 out of the 1 (of the balls in the urn)
3)
left the ball 10 under the carpet

Conclusio:
we have *two* objects marked "10", 20 and so on

Does this solve it

[Wonko the Sane]

No, that doesn't quite solve it because it's assumed in the puzzle that the sorcerer can't see balls under the rug. So as far as he's concerned the only problem is that the 1 is missing. However the puzzle doesn't define if the sorcerer is able to see that. That's a big problem with this is the vague wording. I'd suggest waiting for the solution. I, like I'm sure most others, am tired of arguing moot points and poor declarations.

[new_on_site]

wonki i agree with you, that the phrasing is maybe not good enough to give a math equation. as english is not my mothertounge i didn't put every word on the scale. I just tried my best to anwer the question
> besides ... _could he tell that_ ...
and imho he could, if he just compared the numbers below the carpet against the ones in the urn. btw there is a good clue that those poor demons really worked to the limes
> ... infinite lumps in it
and, allthough not possible in reality
> ... they had no difficulty with this seemingly impossible task
so to me this seems more a linguistic than a math puzzle. let's see
cheers 2 all enjoying this

[Rems]

Hehe, if an infinite number people are replying an infinite number of times, how many times does the ad at the top of the page get loaded?

Minotaur: You rock. I've been reading and trying (unsuccessfully) to solve puzzles here for a couple of years. Keep up the good work! One of my faves is that character, Skinny Dakota.

Rems

[Ghost Post]

I assume that most of you are dead-tired of this puzzle but I believe that I have a point on 'unspecified balls' inside the cheating Demon's Urn (by the way I'm 99.99999999% sure that Minotaur meant Demon#2 to be cheating).

After midnight the Urn will be full of balls with numbers

10000000000000.....zeros to infinity
20000000000000.....zeros to infinity
30000000000000.....zeros to infinity
.
.
.
1100000000000......zeros to infinity
1200000000000......zeros to infinity
.
.
.
4444421000000......zeros to infinity
.
.
.
.
etc

Do you see my point? Although the Demon wrote infnite number of zeros on the ball numbered origially as ball #1 HE NEVERED REMOVED ORIGINAL NUMBER AS WELL AS DIDN'T WRITE ANYTHING OTHER THAN ZEROS. Since there are no two identical original numbers therefore each ball will be unique.

The ball with orignial number two hundred twenty two

22200000000000000......zeros to infinity

will be the ONLY ONE with number two hundred twenty two in the beginning and all other numbers as zeros. If you agree with me on the assumption that the numbers on the balls are infinitely small, although very clearly visible both for Demons and sorceror, and it is possible in this puzzle to write infinite number of infinitely small zeros on the ball (and I don't really see why you shouldn't accept it since you don't have a problem with accepting unfinitely large carpet, as well as unfinitely deep Urn, as well as the fact that the DEMON FINISHED THE TASK) you shouldn't have any problem with my view.

[Tom]

Well, yeah, I see what you're saying .. that's what I thought happened. Is this referring to something someone else said?

[dave10000]

In case anyone is STILL interested in this puzzle, it comes directly out of the rec.puzzles FAQ, where there is a reasonably detailed discussion:
http://www.faqs.org/faqs/puzzles/archive/logic/part5/

Some of the things that the FAQ makes clear are:

(1) The type of task performed by the demons -- an infinite number of steps in a finite amount of time -- is called a "supertask" in the literature.

(2) Standard mathematics does not permit the completion of supertasks -- to analyze a supertask after completion requires an extension of standard mathematics, and THIS CAN BE DONE IN MORE THAN ONE WAY.

(3) By postulating a "kinematic" theory of continuity (which the article defines), one reaches the result of zero balls at midnight for non-cheating Demon 2 -- but by adopting different postulates, one can reach different results, including infinity or, for that matter, 7 (although adopting a postulate that leaves 7 balls in the bucket is not particularly satisfying or useful).

This last point is what I alluded to in an earlier post -- by adopting different postulates (most particularly, different versions of Peano's "parallel postulate"), you wind up with different geometries and different answers to the same geometrical question. Thus, "What do the angles in a triangle sum to?" is not a sufficiently definite question, since different geometries will give different answers. "What do the angles in a triangle sum to in Euclidean geometry?" is well-formed, and the answer is 180 degrees. IN THE SAME WAY the demon's task is not well defined, since we MUST extend standard math to get an answer, and there are different ways of doing so while still being consistent. But asking "What will the Demon's bucket look like at midnight using the "kinematic continuity" extension of standard math?", then the answer is zero. Similarly, asking "What will the Demon's bucket look like at midnight using the "limit ball" extension of standard math?" (which is discussed briefly in the article), the answer is "an infinite number."

A very instructive article about supertasks, which unfortunately does not directly address the bucket problem, can be found at:
http://plato.stanford.edu/entries/spacetime-supertasks/

----

Prediction:

I suspect that the Minotaur will say "zero" balls for the non-cheating Demon 2, and "infinite balls with weird infinitely long non-numbers on them" for the cheating Demon 2. If that is the extent of the Minotaur's answer, that would be unfortunate, since it is a "correct" answer in only ONE of the possible extensions of standard math that is required to solve the problem. If it is the only answer that the Minotaur gives, then he was operating under the "kinematic continuity" postulate WITHOUT SAYING HE WAS DOING SO. That would be like him asking what the angles of a triangle sum to, us saying "180 degrees", and him saying "Wrong! I was operating under Reimann geometry, not Euclidean."

Putting a final, finer point on some of my earlier posts in this thread -- the problem, as asked, is ambiguous, because SOME extension of standard mathematics is needed to solve it, but different (acceptable) extensions lead to different (acceptable) answers. Thus, if the Minotaur gives a specific answer as being correct, that may be true in the extension that he decided to use, but it certainly is not the only answer possible to the problem since no particular extension was specifed.

Thanks for listening.

[Tom]

I'd be inclined to say the situation is a bit more like the Minotaur asking how many degrees there are in a triangle, and you saying "I don't know"; but that's just being picky.

I suppose to support my view, the "kinematical principle of continuity" (which, by the way, just says if you put a ball somewhere before 12:00 and no one moves it, it is in the same place after 12:00) seems to me the obvious thing to assume. But that just depends on what you find paradoxical, and what "strange but true".

Oh well, the puzzle got me thinking, whatever.

[Ghost Post]

As might be expected, I agree.

The stated "kinematical principle of continuity" (which leads to an empty urn for non-cheating demon #2) seems to be pretty intuitive and obvious: you put a ball somewhere and it's there until it's moved. If you placed it somewhere, and it's not there now, it must have been moved.

Note that the rec.puzzles explanation says:

[dave10000]

quote:

---

(extro)
So they demonstrate that the model which leads to an empty urn is consistent. They do not demonstrate anything about other models leading to other conclusions.

---

The article doesn't "demonstrate" it, but it certainly states it in its conclusion:

quote:

---

(FAQ)
I conclude by stating that the result of the super task depends on how
our standard models are enlarged to include the execution of
supertasks. We have given one extension which leads to consistent
results for the supertasks suggested by Ross. Other models may lead to
different, but also consistent, conclusions.

---

And THAT, in a nutshell, was the fundamental point I was making throughout my posts.

Indeed, I AGREE that the "kinematic theory of continuity" is, in some sense, the most "intuitive" way (to me) of extending standard math to account for the Demons' particular supertask. My point has been that it's not the ONLY way, and someone else might find another extension more "intuitive" to him or her.

And finally, replying to Tom:

quote:

---

(Tom)
I'd be inclined to say the situation is a bit more like the Minotaur asking how many degrees there are in a triangle, and you saying "I don't know"; but that's just being picky.

---

I wouldn't say "I don't know." I'd say "Depends on whether you're using Euclidean, Reimann, or spherical geometry, because the answer would be different for each." And I'd be right!

[Tom]

It's very tempting to try to show some inconsistencies with the limit ball model .. no idea how you'd even start doing that though.

By the way, in my last post I meant that I was being picky, not you, dave10000. You would indeed be right to say that. (By the way, you weren't that person who concluded that there is at least one pig in China, and at least one side of it is black, are you? )

[Ghost Post]

When I said "So they demonstrate that the model which leads to an empty urn is consistent" I was only paraphrasing their claim "We have only shown that for these experiments, in our model, we come to consistent conclusions". (assuming 'demonstrate' and 'show' to be roughly equivalent)

[dave10000]

extro:

I have no problem with (and I agree with) your statement that "So they demonstrate that the model which leads to an empty urn is consistent".

What I took issue with in my earlier post was your NEXT sentence that "They do not demonstrate anything about other models leading to other conclusions." I just wanted to point out (in my earlier post) that, while the article does not "demonstrate" that other models lead to other conclusions with the same depth that the model examines the "kinematic" theory, the article certainly SAYS that other models may lead to different, but also consistent, conclusions. Indeed, that conclusion is reached by the very last sentence of the article. And the existence of different solutions has been my point all along.

[Tom]

I thought a bit more about these "limit balls", as they call them.

I suppose if you consider the urn to be an infinitely tall cylinder, just wide enough for one ball. If you had an opening at the bottom, (a bit like this)

then you could consider demon #2's task to be -
Put balls 1-10 in the top of the urn, take the bottom ball out. Put balls 11-20 in the urn, take the bottom ball out. Continue. (I'm ignoring the problem of putting the balls in the top of an infinitely high urn .. you can get round this, it doesn't matter.)

In this case, you could consider the principle "If a place in the urn is always filled from some time up until 12:00, then it is filled at 12:00" to be fairly reasonable, I suppose; this would give you an infinitely full urn.

I can't see why this wouldn't be consistent (bit contrived though).

[This message has been edited by Tom (edited 05-26-2000).]

[Tom]

I just thought of this (while out buying some lunch).

Suppose, to get round the infinite urn problem, we decide to have ball 1 of mass 1/2, ball 2 of mass 1/4, .. ball n of mass 1/2^n. The all the balls together will have mass 1.

If you use the "balls stay where you put them rule", then all the balls get taken out, as before, giving an empty urn.

Now, suppose you use the principle from the last post, that a place stays filled at 12:00 if it is filled up to 12:00. As before, you get an infinite number of balls in the urn. But how much is in the urn, masswise? Well, we have taken out mass 1 of balls, obviously. And all the balls make up a mass of 1, so there is a mass of 1-1=0, in the urn (no infinity-infinty=? problem here). So there are an infinite number of balls in the urn, which together have mass 0. So each one has mass 0.

All sounds a bit fishy to me.

[Wonko the Sane]

Look, we're probably all right. The Minotaur said that this puzzle is a paradox. There's probably more than one right answer. I can see merit to a lot of the arguments, so I'm not taking sides anymore. Tom, you make a valid point, but although there is a spot that is always filled, you still remove every ball you add. So depending on interpretation, you may or may not be right. I'm just gonna wait to find out though. Still, thanks that post Tom, it was a nice spin on the same old points.

[Ghost Post]

Maybe the idea behind the cheating demon #2 was to have it be like the non-cheating demon #2, but using the "limit ball" model. After all (playing devils advocate here, perhaps), what should be so different between:
A) placing 1 through 9 in the urn, and painting a zero on the 1 ball to make it a 10
B) placing 1 through 9 in the urn, and actually replacing the 1 ball with the 10 ball (all nine places occupied by the nine balls remain occupied after the replacement - and this would hold for all remaining sets of nine balls placed in)

[CrystyB]

Tom, it's a different ball in the spot you selected. The original ball HAS moved, so it's not in the same place. It will be outside. And i love wour way of dealing with inf-inf as with 1-1

Also, for ZenBeam: NOW (today, 28th of May 2000) it is clear to see that the Minotaur DID fix the problem. Perhaps he had fixed it a longer time ago, but has delayed updating the site. Anyway, it's done now!

[This message has been edited by CrystyB (edited 05-27-2000).]

[new-on-site]

Hello again, and thanks to giving me the opportunity to learn about the aleph-0 set (and all the others). In light of this concept, there is indeed a difference, if I understood the concept correctly.

Daemon 1 will produce by his task one set of infinite number of balls inside the urn, and one infinite set of balls outside (the ones he had to take away) - but they are still of the same _cardinality_ (both aleph-0, like the natural numbers).

Daemon 2 would produce by his task one set of infinite number of balls inside his urn, but a set of different cardinality outside the urn (=under the carpet or whereever), because there is no 1:1 relation between the two sets, i.e. second move remove number 2 is still from the first set of balls given to the urn.

By doing the cheat, he ends up again with an aleph-0 set because now there's again a 1:1 relation between the in's and outs.

It would be interesting for me to find out _which_ cardinality the should-be set of daemon 2 was - I am not experienced enough with this stuff.

For an interesting article about the many faces of "infinity" you might check http://www.mathacademy.com/platonic_realms/minitext/infinity.html

Have fun & surprise

[Ghost Post]

No, the sorceror couldn't tell.
If the demon had been following instructions, he would have taken out one ball for every ten he put in: 1 for 1-10, 2 for 11-20, etc. His way, he never took balls out(just changed them and put them back in), and put in nine every time. I think this translates, number-wise, into the same thing.

[New User]

Firstly...nice puzzle and posts made for good reading. I think the demon is lazy since by this scheme he eliminates some operations.

Consider

prescribed way:
put in 10, search for 1, take out 1.
repeat.

Lazy way:
put in 9, search for 1, paint 1.

assuming puting 1 under the carpet is less work than puting it into the urn and painting 1 is easier than taking it out. And remembering that this small difference in effort has to be repeated uhhhhh...infinite times, it kind of makes sense that the demon 2 would try to optimize.

But I do not think that is the point.

By just looking inside the urn, the sorcerer will not be able to tell that the demon has cheated, however all he has to do is ask the demon to show him those that had been removed, they will be 10*x where x would be the expected series of numbers outside.

The number of marbles will remain unchanged due to the loops being repeated an equal number of times.

well...really looking forward to seeing the solution posted by the minatour.

[stoatboy]

maybe this already occurred to everyone else, but it now appears to me as if the way the Minotaur has posed the question is exactly the dichotomy between the limit-ball versus kinematic continuity. We've been couching syntactically with the argument that 'x ball is removed' versus 'x ball is never removed', but the cheating daemon #2 provides us with an obvious limit ball mapping: a spot in the urn is taken up with ball 1, then with ball 10, then with ball 100, etc...and, in fact, it's the same ball, with a new number painted on it. The Minotaur seems to be guiding us towards kinematic continuity in the non-cheating case and limit-ball continuity in the cheating case. We just weren't seeing it.

Clever bastard, that Minotaur.

[dave10000]

KUDOS to the Minotaur for a spot-on answer to the "Urns of Infinity" puzzle! I'm glad he did not take the "easy" way out by saying the answer had to be one way or another.

Could he possibly have been influenced by some of the near-200 posts on this board???

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Eternity is a long time; especially the end.
-- Niels Bohr