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Topic: Re: Chew on This (Read 5588 times) |
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ThudnBlunder
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Re: Impish Pixie
« Reply #25 on: Apr 7th, 2003, 6:53pm » |
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Yes, I was going to say that, although it is quite easy to accept - or visualize - that the limit of f(x) as x->a is not necessarily equal to f(a), in this (discrete) case the discontinuity is at infinity - making it rather difficult to visualize, much less accept.
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| « Last Edit: Apr 10th, 2003, 10:08am by ThudnBlunder » |
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Icarus
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Re: Impish Pixie
« Reply #26 on: Apr 7th, 2003, 7:35pm » |
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So - Bring it in from infinity! Instead of labeling the sets B with integers, label them with egyptian fractions: The symbol representing the nth set is B1/n. The final set is B0 The limit may be stated as:
limx[to]0 Card(Bx) = [smiley=varaleph.gif]0
while Card(B0) = 0, (with the understanding that x is restricted to numbers of the form 1/n, n [in] [bbn]).
This is completely equivalent to the original statement. When it comes to limits, infinity is no different than ordinary numbers. If you add [infty] and -[infty] to the real line, one at each end, then what you get is topologically equivalent to a closed interval, such as [-1, 1] ("topologically equivalent" means that it behaves the same as far as limits and continuity are concerned). You have to get into higher infinities before topological behaviour gets weird.
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| « Last Edit: Aug 18th, 2003, 5:47pm by Icarus » |
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Rujith de Silva
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Re: Impish Pixie
« Reply #27 on: Apr 9th, 2003, 11:58am » |
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I think intuition fails us here because asking "What is the state
after an infinite time?" is radically different from asking "What is
the state after a really long time?" Similarly, saying "Dog created
the universe an infinitely long time ago" is equivalent to saying "The
universe has always existed."
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Chronos
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Re: Impish Pixie
« Reply #28 on: Apr 9th, 2003, 7:46pm » |
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Actually, my claim is that the question is meaningless. Once you accept that there exist similar questions which are meaningless (like whether the light bulb is on or off), then you have a bit more of the burden of proof, to show that this question is, in fact, meaningful. I can think of two ways to model this question: By reality, or by mathematics. But I can't model it by anything in reality, because nothing can move balls around that fast. And on the other hand, I've never seen a precise mathematical way of expressing this question. The closest I can come is by stating a function that gives the state of the basket (and the balls in it) as a function of time. But as described in the problem, this function only has a domain of (-oo, +1 hour). If you ask me what the value of this function is at a point greater than or equal to one hour, I have to extend the function in some way. But heck, I can think of a heck of a lot (2^(2^Aleph0), to be precise) different ways of extending it, to give any answer you like. That's no good.
So, how does one express this problem in a mathematically precise way?
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Icarus
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Re: Impish Pixie
« Reply #29 on: Apr 9th, 2003, 9:27pm » |
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The way wowbagger and I have already expressed it is quite precise mathematically.
For each k, the location of ball k at the end of the process is easily determined.
Ball k was put in the basket early in the process, and removed a short time later. After this, ball k was never touched again. To claim that the question of its location after the process ends is "meaningless" is rather incredible! Ball k changes its position twice. Its position for every time after its second move is the same, including times after the process is finished.
That position is "outside the basket" for every k.
Presuming that the only balls available to be in the basket are those meantioned in the puzzle, it is clear that the basket must be empty at the finish, for the position of every ball at that time is "outside the basket".
As I stated in my previous posts, your light switch example is NOT similar to this one. That example has an object jumping between two discrete states infinitely many times. Its final state cannot be determined.
This puzzle does NOT have anything necessary to it which jumps infinitely many times between discrete states. Not even the set of balls in the basket does this, because the sets are not truly discrete: they converge to a limit set: [emptyset].
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| « Last Edit: Aug 18th, 2003, 5:49pm by Icarus » |
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BNC
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Re: Impish Pixie
« Reply #30 on: Apr 10th, 2003, 12:12am » |
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on Apr 9th, 2003, 11:58am, Rujith de Silva wrote:
...Similarly, saying "Dog created
the universe an infinitely long time ago" is ...
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Sounds like an interesting religion 
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Evenstar
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Re: Impish Pixie
« Reply #31 on: Apr 10th, 2003, 4:03am » |
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on Apr 2nd, 2003, 10:53am, THUDandBLUNDER wrote:| QUESTION: After an infinite amount of time has elapsed, how many balls are in the bucket? |
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I can't help thinking "After an infinite amount of time has elapsed"... How can there be anything after an infinite amount of time? At what point would that be?
Icarus also talks about "when the process is finished". But given that it is infite, how can it be finished?
Anybody patiently enough willing to explain that?
Kind regards,
The Evenstar.
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Rujith de Silva
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Re: Impish Pixie
« Reply #32 on: Apr 10th, 2003, 5:43am » |
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on Apr 10th, 2003, 4:03am, Evenstar wrote: I can't help thinking "After an infinite amount of time has elapsed"... How can there be anything after an infinite amount of time?
Anybody patiently enough willing to explain that?
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Yes, but it will take an infinite length of time to explain!
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| « Last Edit: Apr 10th, 2003, 5:44am by Rujith de Silva » |
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ThudnBlunder
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Re: Impish Pixie
« Reply #33 on: Apr 10th, 2003, 10:13am » |
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Quote:If the phrase 'after an infinite amount of time has elapsed' bothers you, then we can change the problem so that the 1st put-in-and-take-out operation is completed in 1/2 minute, the 2nd operation is completed in 1/4 minute, the 3rd in 1/8 minute, and so on.
Now you can ask the question after 60 seconds, and "infinite time" is not longer an issue. |
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Evenstar, if you can't be bothered reading the question properly then please don't waste people's time by replying.
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| « Last Edit: Apr 10th, 2003, 10:14am by ThudnBlunder » |
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Re: Impish Pixie
« Reply #34 on: Apr 10th, 2003, 10:21am » |
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on Apr 10th, 2003, 12:12am, BNC wrote:
Sounds like an interesting religion
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Rujith is obviously an agnostic dyslexic insomniac, lying awake at night wondering if there is a Dog.
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Evenstar
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Re: Impish Pixie
« Reply #35 on: Apr 11th, 2003, 1:14am » |
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on Apr 10th, 2003, 10:13am, THUDandBLUNDER wrote:
Evenstar, if you can't be bothered reading the question properly then please don't waste people's time by replying.
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Yes, you have a point there. I had read the question completely, but between the time the question was asked and the time the answers started comming in, my mind had left out that part. My apologies for that.
Maybe it's because I have this mental block which prevents me from trying to cram an infinite amount of things into a finite container. Then again, I never had a problem with 1/2 + 1/4 + 1/8 + 1/16 + ... = 1, which is pure math. Transforming them from numbers to slices of time I can grasp, attaching an action to the timeslice is what makes it difficult for me.
Mind you, it's not that I doubt that the reasoning behind it is wrong. The proof is quite, hmmm, how could I put it in english... it's quite sound.
It's just me who can't wrap my brain around the idea that had the pixie taken out the highest numbered ball instead of the lowest, the bag would hold an infinite amount of balls instead of none. The two actions (removing the lowest/highest numbered ball) seem identical yet have a totally different outcome.
Kind regards,
The Evenstar
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ThudnBlunder
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Re: Impish Pixie
« Reply #36 on: Apr 11th, 2003, 4:02am » |
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Quote:It's just me who can't wrap my brain around the idea that had the pixie taken out the highest numbered ball instead of the lowest, the bag would hold an infinite amount of balls instead of none. The two actions (removing the lowest/highest numbered ball) seem identical yet have a totally different outcome.
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Yes, it is really counter-intuitive, isn't it? 
How about Simpson's Paradox for a really easy to understand yet counter-intuitive result?
(Sorry for being so brusque earlier.)
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| « Last Edit: Apr 14th, 2003, 2:37am by ThudnBlunder » |
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Jacky Poon
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Re: Impish Pixie
« Reply #37 on: Apr 13th, 2003, 6:55pm » |
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At first I thought argument 2 would be correct, but that removing the highest numbered ball instead would alter the result is making me uneasy.
Consider the following scenario:
You have an infinite number of small balls, only they are numbered from 1 upwards once in pen and in pencil so that the pencil numbers (which are changable) and pen numbers (permanent) coincide.
Every minute you throw in the two balls as in the question. But every minute:
1. The Pixie swaps the pencil numbers around so the lowest numbered ball in pen has the highest number in pencil.
2. The Pixie removes that ball. (Pen: smallest, Pencil largest)
QUESTION:
After an infinite number of time how many balls are there?
1. Looking only at the pen numbers this is equivalent to the original question, each time the pixie is removing the lowest value of the ball. Therefore according to the there is 0 balls left.
2. Looking at the number of pencil numbers that are written on the remaining balls - this is equivalent to removing the highest pencil number each round, so there should be infinite pencil no's written on the balls. Since there is only ever 1 pencil number per ball, there are infinite balls.
But infinite balls does not equal no balls. Therefore either
a) there is a flaw in this argument
or
b) there is a flaw in 0 as a solution to the original problem.
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Icarus
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Re: Impish Pixie
« Reply #38 on: Apr 13th, 2003, 7:28pm » |
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The answer is (a) there is a flaw in the argument.
You shouldn't be looking at the markings, pen or pencil, at all. You have to look at the BALLS.
With or without the pen markings, with or without the pencil markings, the basket is empty - because each ball was put in once, and was removed once. From there on out, it is not in the basket. When the infinite process is finished, each ball must be outside the basket, so the basket must be empty.
What will happen in your scenario in the end is that
(1) the basket is empty.
(2) each ball, outside the basket will have 2 numbers written on it, with the number in pencil being twice that in ink.
The outcome of this process is not controlled by the numbers, and the puzzle could have been stated without numbers at all:
You have a countably infinite number of balls (this means there are no more balls than there are natural numbers). You put them into the basket two at a time. The imp removes the ball (or one of the two balls) that have been in the basket longest.
Stated as such, the basket will still be empty, because every ball is eventually removed and never touched again.
Change the imp's behavior to: "the imp removes one of the two balls you just put in." Then, the basket will contain infinitely many balls at the end, because out of every two balls you put in, one of them is never removed.
Change the imp's behavior to: "For the first 12,642,970 times you put two balls in, the imp removes a ball at random. After this, the imp ignores the balls still in the basket, but removes the ball that has been in the basket longest of all those placed in after this point." - In this case at the end, the basket will contain 12,642,970 balls - namely the ones put and never removed before the behavior change.
Change the imp's behavior to "The imp removes a ball completely at random", and you do not have enough information to determine the result completely. You can say that the probability of the basket being empty is 100% (with infinitely many cases, this is not the same as saying that the basket must be empty). - I read this result in another discussion of this puzzle, and have not checked myself if it is true, but it sounds right.
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mypalmike
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Re: Impish Pixie
« Reply #39 on: Apr 24th, 2003, 1:03am » |
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Suppose we do refine the question to involve the pixie taking out the "oldest ball". Consider the alternate timing suggestion, where the timing between successive events is halved as t approaches 60 seconds. At the "end of the process", the time slice is exactly zero. Which ball is oldest when all balls are the same age? Arguably, the process can not be brought to a conclusion because the pixie can not identify the oldest ball.
This is not simply playing with semantics or words. In the original question, the set theory explanation assumes that the lowest numbered ball can be identified through completion of the process. Is this a valid assumption?
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Icarus
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Re: Impish Pixie
« Reply #40 on: Apr 24th, 2003, 5:50pm » |
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The time slice between any two steps is never actually zero. By the time it gets to zero, the process is finished, so the imp does not have to worry about choosing which ball is oldest. There are no balls left to choose between.
There is an interesting discussion about a variant of the same puzzle at this site, - It's the third puzzle on the page."
In particular, he mentions an "alternative model" that in my opinion is just barely at the borders of being reasonable/unreasonable.
To be fair, though, I should say that the solution given here is unique only under the following assumptions:
1) The only things that could possibly be in the basket at any time are the balls mentioned in the puzzle.
2) If a ball is not moved at all after some point in time, then its position at that time will also be its position at the finish.
Both of these assumptions seem reasonable enough to me to be taken for granted. If we were to allow violation of them here, then why should we not allow similar violations in other puzzles? ("He can't know what his hat color is, because the judge was hiding a chartreuse hat behind his back!" "None of the prisoners can figure out their hat colors because the hats sponeously rearrange themselves between guesses.")
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rmsgrey
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Re: Impish Pixie
« Reply #41 on: May 14th, 2003, 2:21pm » |
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I'm still not convinced that the canonical solution (which I accept as consistent with accepted mathematics and generally useful) is the only consistent way of modelling this type of situation - my intuitive answer is that at time oo there are oo many balls left inside, labelled oo+1 to 2oo...
Going back to the original post, the second argument (what is the lowest numbered ball in the bucket) is susceptible to the counter: "what is the number of the last ball put into/taken out of the bucket?"
Looking at Icarus' last post in more detail, it seems that, to support his solution's uniqueness, his first assumption should be that only balls with finite labels are involved - the original statement of the problem only says that there are an infinite number of uniquely numbered balls available. Whether they are restricted to natural numbers only or allow countable transfinite numbers is not made clear. Of course, changing his assumption this way means that it is no longer a special case of a universal underlying assumption for "nice" riddles (as opposed to "joke" riddles or "bad" riddles).
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wowbagger
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Re: Impish Pixie
« Reply #42 on: May 15th, 2003, 2:30am » |
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on May 14th, 2003, 2:21pm, rmsgrey wrote:| my intuitive answer is that at time oo there are oo many balls left inside, labelled oo+1 to 2oo... |
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And at what time was ball number oo+1 put into the bucket?
Quote:| Going back to the original post, the second argument (what is the lowest numbered ball in the bucket) is susceptible to the counter: "what is the number of the last ball put into/taken out of the bucket?" |
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That's like asking: "What is the largest natural number?"
Hm, the naturals are not bounded above...
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rmsgrey
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Re: Impish Pixie
« Reply #43 on: May 15th, 2003, 1:07pm » |
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on May 15th, 2003, 2:30am, wowbagger wrote:
And at what time was ball number oo+1 put into the bucket?
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At time 1+oo/2. I don't see a problem. As I said, this isn't the standard extension of numbers to infinity. I should probably say that I'm using oo to indicate a fixed (unspecified) countable infinity rather than the unique countable infinity of the standard version. I'm also not certain how useful this alternate model is, or even if it can be formalised in a consistent fashion. If it can, then there seems to be no reason why "realistic" processes should prefer one model of infinity to any other since there is no direct experience of infinity in real life.
Quote:
That's like asking: "What is the largest natural number?"
Hm, the naturals are not bounded above...
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So there is no last ball moved either into or out of the bucket? In that case either no balls are ever moved, the process can never end, or the argument that there are no balls in the bucket because you can't name the smallest is seriously flawed.
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rmsgrey
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Re: Impish Pixie
« Reply #44 on: May 15th, 2003, 1:23pm » |
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on Apr 4th, 2003, 12:52pm, cho wrote:[...]has anybody heard the news about a problem with quantum theories. If space and time are quantized, then space should be like a foam and photons passing through it will take different paths and slightly different speeds. That should make light traveling vast distances blurry, but Hubble photos of galaxies 5 billion light years away are razor sharp. So no quanta. If no quanta then time and space can be divided into infinitely small units. That should mean that the Big Bang began with an infinitely hot universe in an infinitely small space. Not possible.
So we'd better figure out this infinity quickly or the whole universe may be an impossibility and cease to exist. |
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The quantized foam would be on a scale of 10-33m while the distant galaxies are only around 1026m away and (if I remember correctly) have a size of about 1020m or an initial granularity of one in 1053 or so. For fuzziness to be visible, the accumulated error would have to come to about one part in 104 and even then, that only represents a hairs-breadth of error over a canvas one meter across... I'm more impressed that the optical instruments on Hubble can resolve an object roughly a million times as far away as it is large with such clarity than that planck length effects don't distort the image.
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ThudnBlunder
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Re: Impish Pixie
« Reply #45 on: May 15th, 2003, 10:43pm » |
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Quote:| At time 1+oo/2. I don't see a problem. As I said, this isn't the standard extension of numbers to infinity. I should probably say that I'm using oo to indicate a fixed (unspecified) countable infinity rather than the unique countable infinity of the standard version. |
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rmsgrey, if there are an infinite number of infinities and you invent another, how many infinities do we now have?
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rmsgrey
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Re: Impish Pixie
« Reply #46 on: May 16th, 2003, 11:55am » |
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on May 15th, 2003, 10:43pm, THUDandBLUNDER wrote:
rmsgrey, if there are an infinite number of infinities and you invent another, how many infinities do we now have? |
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Depends on your model of infinity. The standard model takes the continuum hypothesis (that the smallest infinity greater than aleph-null is aleph-one) as an axiom, but it has been shown that the continuum hypothesis is independent of the other axioms in the standard model (as with the parallel postulate in geometry). If you work within the standard model, then, if I remember correctly, there are only countably many infinities (formed inductively by diagonalisation for example) so as soon as you get an infinite number of them, you can no longer introduce new ones. If you assume the negation of the continuum hypothesis, then it depends on the properties of the lowest infinity greater than aleph-null, and the number of infinities could remain countable or become uncountable. Though, of course, you're not really making up a new infinity in that case, simply changing the rules so that the "new" infinity (and all its consequences) have "always" been there.
The question of "inventing" a new infinity is misleading - and ambiguously phrased - if you have an infinity of infinities, which of them is the cardinality of the set of infinities? The only possible answer to your question is, of course, "an infinite number", but not necessarily the same infinite number.
If this debate goes on much further, I'll have to go back to the axioms and see what sort of system I can construct (which will probably keep me quiet for a while). The major advantage of the standard model is that any time a countable infinity comes up, it's guaranteed to be the same aleph-null (give or take a minus sign). For my non-standard model, it looks like you can only compare two countable infinities if you can compare their generating processes - pick one to be your reference and describe the other(s) relative to it.
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ThudnBlunder
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Re: Impish Pixie
« Reply #47 on: May 17th, 2003, 7:43am » |
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Quote:| At time 1+oo/2. I don't see a problem. As I said, this isn't the standard extension of numbers to infinity. I should probably say that I'm using oo to indicate a fixed (unspecified) countable infinity rather than the unique countable infinity of the standard version. |
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rmsgrey, you talk the talk, but....
....as this puzzle does not involve uncountable infinities, I fail to see what it has to do with CH.
Furthermore, you want "oo to indicate a fixed (unspecified) countable infinity".
Isn't the onus on you to 'specify' exactly what you are referring to? (And are there any unfixed infinities?)
Once you have done that, I will try to take half-seriously your answer of '(1 + 00)/2'.
Perhaps the first axiom of your system is that 00 is an odd number? A very odd number.
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| « Last Edit: May 17th, 2003, 8:37am by ThudnBlunder » |
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Icarus
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Re: Impish Pixie
« Reply #48 on: May 17th, 2003, 9:28am » |
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Looking carefully at the wording of the puzzle as originally stated, I have to agree with rmsgrey that it does indeed allow for the possibility of non-natural number labels on the balls, and since the stop time is indicated as "after an infinite amount of time", it does not imply that only finitely counted steps were taken. So it is indeed possible for infinitely labeled balls to come into play.
This could be fixed with one or two small rewordings of the puzzle. But since that shuts the discussion down, instead let's see what we can come up with for the puzzle "as is", along with my two conditions given in my previous post, also "as is".
I have said before in other threads that there are all sorts of infinities, and you can define new ones at need - provided of course that your definitions are consistent. There are some conditions on the infinities allowed here, though. In particular, since our process takes place in time, it has to be totally ordered, and therefore the balls affected by it must also have ordered labels. (There can be all sorts of balls not in this order, but they will never be manipulated by any step in the process. So by the second of my conditions, they won't ever be in the basket). This still leave things open to multiple types of infinite numbers. I will have to consider it longer to see what I come up with.
I do have some comments to make on this, however:
Quote:| Depends on your model of infinity. The standard model takes the continuum hypothesis (that the smallest infinity greater than aleph-null is aleph-one) as an axiom, but it has been shown that the continuum hypothesis is independent of the other axioms in the standard model (as with the parallel postulate in geometry). |
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I was not aware there was a "standard model" of infinities. The are three common types (or models) of infinite numbers, and again, others can be defined. The common ones are Cardinals - which appear to be the ones you are refering to by this, Ordinals - you might want to consider them as labels, and of course the continuum infinities at the ends real line or the complex plane.
Assuming that you mean Cardinals - they do not depend on the Continuum hypothesis. And for most results it is not needed. The Continuum hypothesis merely asserts that the cardinality of the real numbers is the second infinite cardinal (it must be at least the second). The generalized continuum hypothesis says that for any infinite set A, If P(A) is the set of all subsets of A, then Card(P(A)) is the next cardinal higher than Card(A). This hypothesis is useful in relating cardinals with sets, but is not particularly important in the existance and behavior of cardinals themselves.
Quote:| If you work within the standard model, then, if I remember correctly, there are only countably many infinities (formed inductively by diagonalisation for example) so as soon as you get an infinite number of them, you can no longer introduce new ones. |
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Sorry, but you do not remember correctly. Maybe Russell-Whitehead style set theory can handle the concept of the cardinality of all cardinals (I don't know - I've never studied cardinality in that system), but in the more commonly used Zermelo-Frankel approach, the is no such thing as the set of all cardinals (to have such a thing introduces paradoxes). And without that set you cannot define how many cardinals there are. But while you can't get a full set of cardinals, you can get a set of all cardinals less than a particular one, and - if I recall correctly - some of these sets are uncountable. A result which does not depend on the continuum hypothesis.
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Impish Pixie
« Reply #49 on: May 17th, 2003, 12:10pm » |
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Quote:Looking carefully at the wording of the puzzle as originally stated, I have to agree with rmsgrey that it does indeed allow for the possibility of non-natural number labels on the balls, and since the stop time is indicated as "after an infinite amount of time", it does not imply that only finitely counted steps were taken. So it is indeed possible for infinitely labeled balls to come into play.
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Icarus, I don't understand how that can be, as there are an INTEGRAL number of operations - unless you are
likening "an infinite amount of time" to the real line? But even the real line is punctuated by integers, just as
"an infinite amount of time" is punctuated by one-minute intervals. Or perhaps it is possible to interpret the phrase to mean (for example) "after alephn units of time"?
In all the threads and sub-threads that I have read about this puzzle, I have never seen any claims that we may be using an uncountable number of balls.
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| « Last Edit: May 18th, 2003, 6:51am by ThudnBlunder » |
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THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
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