Search found 79 matches

Wed Aug 01, 2018 9:35 am UTC
Forum: Logic Puzzles
Topic: Where are all the balls?
Replies: 15
Views: 6580

Re: Where are all the balls?

This is why I thought it worth discussing - so far we've had three types of resolution: ⋅  The problem is ill-posed, you can't talk about the end-state of a never-ending sequence ⋅  The pots are all empty, therefore all the balls have vanished ⋅  The balls end up in pot...
Mon Jul 30, 2018 1:02 pm UTC
Forum: Logic Puzzles
Topic: Where are all the balls?
Replies: 15
Views: 6580

Where are all the balls?

This is somewhat related to the Ross-Littlewood (balls and jugs) paradox, but I feel it's different enough to merit some discussion. Suppose I have an infinite number of pots, labelled p 1 ,p 2 ,p 3 ,... initially empty but each capable of holding one of an infinite number of balls, labelled b 1 ,b ...
Fri Dec 23, 2016 2:59 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

I think I've just realized what your issue may be. You're (perhaps subconsciously) taking the end of the supertask to be after some vaguely-defined infinite number of steps - of first halves of TCns, if you will, both finite and infinite - when in actuality the end of the supertask is merely after ...
Fri Dec 23, 2016 12:52 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

It's amazing this has gone on for so many pages. I think it's worth reiterating how simple the solution really is. For every ball labelled < infinity, some task removes it from the jug and no subsequent task adds it back. There are no balls labelled >= infinity (or any other kind of labelling - rea...
Thu Dec 22, 2016 4:25 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

kryptonaut wrote: Demki wrote: I did not see such a case described, could you describe it for me? Is it that hard to imagine? For each step of the supertask add 10 balls and remove 1. What do you have at midnight? In my point of view, it is underdefined. I am unable to know for any given ball if it...
Thu Dec 22, 2016 1:57 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

In task C there are not "two infinite supertasks running in parallel" but an infinte set of finite subtasks (to be precise, an ω-indexed ordered set of finite subtasks) TC1, TC2, ... all running in parallel. These subtasks are separable because each one acts on exactly one ball, and no tw...
Thu Dec 22, 2016 10:17 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Right (with regard to the last bit; the rest is underdefined). As I have said before: If I watch you add and remove balls, but I can't see the labels, then if you ask me what's in the jug the answer is "I don't know." If you then show me the contents of the jug, the contents that I see wi...
Wed Dec 21, 2016 1:17 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Your problem is your continued insistence that somehow balls like ω and ω+1 and 2ω just magically show up. All the balls we started with had natural numbers on them. So which natural number was originally on the now-lowest ball in the jug, which you claim is now numbered at least ω? When did it acq...
Tue Dec 20, 2016 5:46 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

You seem to be leaning toward my option 3, but you keep talking instead about balls and labels that simply do not and cannot exist. If instead of sides we had colors, what do you think would happen? If each step involves removing the lowest ball and painting the next 10 balls blue, what's left at m...
Tue Dec 20, 2016 4:30 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

There is no last finite-numbered ball, and no ball with label omega (or beyond omega) can suddenly appear since at no step do we change any labels. At midnight, there are only three options if we start with natural-numbered balls: 1) Some remain in the jug. All of them have finite labels because al...
Tue Dec 20, 2016 2:41 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

(C) For task C, imagine that the jug is divided into two halves, L and R, and initially all the balls are in the L half. The finite subtasks are TCn: On step int((n-1)/10)+1 move ball n from L to R. On step n take ball n (which will be in side R) out of the jug All other steps: do nothing Firstly, ...
Fri Dec 16, 2016 4:51 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

I've been thinking some more about this (don't groan :) ), particularly in light of mward's post . I imagined an infinitely long train with numbered carriages, passing through a station. At any finite time it's possible to say which carriage is passing through - but after an infinite number of carri...
Tue Dec 13, 2016 10:54 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

I could have defined it to include the endpoints. Then at midnight, there would be exactly one element in the set, and that element would be zero. It would have one element because set theory coalesced an infinite number of zero values into one. That is an indication that set theory is not the righ...
Sun Dec 11, 2016 6:02 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

You claim this is not one of those times. Are you making that claim just because the consequences of it being the natural numbers go against your finite-set-based intuition? Or is there some more concrete reason? I claim there are numbers that take an infinite number of steps to count to, for the v...
Fri Dec 09, 2016 4:02 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

The puzzle starts " Suppose I have infinitely many balls, numbered 1,2,.. and so on. " It says nothing about finite numbers, or natural numbers. Just "infinitely many balls". If you actually count them all , it's no surprise you get an infinite number. If you try to model it usin...
Fri Dec 09, 2016 2:37 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

In the original problem and the original problem only, what mechanism is used to create balls with infinite labels? By starting an infinite task and actually finishing it . We set out to count to infinity and we got there . We didn't just tend towards it , we really did it . The whole thing. Done. ...
Fri Dec 09, 2016 11:21 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Then let's try a different tactic. We are not merely dealing with a generic "fully instantiated infinite set of numbers" - we are dealing with a particular set: the natural numbers. So let's talk about those. How do you define the natural numbers? I ask "why are we dealing with an in...
Thu Dec 08, 2016 11:25 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

A) When we only add balls ten at a time sequentially in order, we end up with exactly N in the jug at midnight. B) If we start with N in the jug, and remove balls one at a time sequentially in order, then we end up with an empty jug at midnight. C) YOUR CLAIM: If we do exactly what we did in (A), o...
Wed Dec 07, 2016 12:37 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Regarding set partitioning: Take an infinite set S ={1,1,1,1,....} Append another 1. Or a hundred of them, or an infinite number. You have the same set. Do that backwards, you have partitioned S into an infinite set and another set. Take an infinite set N ={1,2,3,...} Append an infinite value. Or a...
Mon Dec 05, 2016 9:53 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

If L were merely infinite, then your argument might have some merit. However, L is not merely infinite - it contains ALL the natural numbers. There isn't any mapping going on here. ALL the natural numbers from 1 to where? The only way to compare them is to set up a 1:1 mapping. Why should g be defi...
Mon Dec 05, 2016 1:26 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Ok, jumping in again... Let's skip the labels and the balls and all the distractions. The labels are important, I don't think they should just be dismissed. Now consider the three-partition L-C-R scenario. At the end of the supertask, all the natural numbers are in L. It doesn't matter how it gets t...
Thu Dec 01, 2016 9:28 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Several people spent the better part of five pages asking for them. I think if there were any, they would have been provided. I'll say just this, then I'm gone. If you want to model the game with sets of numbers then you have to accept that a number is a number is a number, whether it's represented...
Wed Nov 30, 2016 4:40 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

But you can put the 10% of the naturals you took away in a bijection with the entire set of naturals, in their usual order. (For example, if you took out 1, 11, 21, etc. you could use 1 -> 1, 11 -> 2, 21 -> 3, etc.) 10% of the naturals is the exact same size as the naturals. And when you try to tak...
Wed Nov 30, 2016 10:25 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Can you prove that it's even possible to divide N this way? And can you prove that the puzzle does force this division? It's not possible to take set N and pick a specific number partway up the set. But it's possible to construct two sets simultaneously, using every member from N , but keeping the ...
Tue Nov 29, 2016 5:23 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

And it's possible that I've missed your explanation on where the infinite numbered balls came from since they did not exist at the start. Would you mind pointing it out to me? They arise because the set N is divided in the way I just described, at a point relative to the size of N . The numbers in ...
Tue Nov 29, 2016 4:39 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

@Poker, @Demki - Ok, I agree you can't have a uniform distribution of N. I was just trying to make a visual metaphor that would show how dividing N at some point relative to the size of N would make numbers with ordinal omega appear.
Tue Nov 29, 2016 4:34 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

The trouble is, what we're dealing with in the original problem is the natural numbers in their usual ascending order. In order for your "infinite numbers" to exist, there needs to be an infinite amount of numbers followed by something. This doesn't happen with the natural numbers in thei...
Tue Nov 29, 2016 3:48 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Well, C follows L (note notation) inasmuch as all elements of C are strictly greater than any element of L. This doesn't force any elements of C to be infinite; It just ensures that at the end of the supertask, C does not contain any finite elements. C could be empty. Elements do get removed from C...
Tue Nov 29, 2016 10:00 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

If you tack a number onto the end of it in an ordered set, {1,2,3,... X}, then by definition that number has to be bigger than any finite natural number. Fine, but at no point do we "tack a number onto the end of {1,2,3...}". We have {C} specified as following L My answer is an infinite n...
Mon Nov 28, 2016 4:49 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

θ is the thing that you change at each step, from which all other values in the puzzle are defined. So why is an the value that drives the puzzle? If we think of the puzzle as a dial that we turn, which points a laser toward the line y=1, then the point of contact of the laser is equivalent to an. ...
Mon Nov 28, 2016 11:07 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

How would you like to resolve this? The value 'driving' the puzzle is a n , which is analogous to n in the jugs game. The angle subtended by a n is zero when n is infinite. But that doesn't mean that the area of the triangle is zero. Zero times infinity can be any variety of things. A point source ...
Sun Nov 27, 2016 4:39 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Ok, let's look closer. With the substitution k=n-1, we can rewrite the example above as: L={1,2,3,4...k} C={k+1} R={k+2, k+3, k+4...} with the supertask of moving the lowest element of R into C, the lowest element of C into L, and incrementing k. This is exactly the same puzzle, is it not? Now, con...
Sun Nov 27, 2016 12:42 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

@ Demki - thanks for the clarification. When I say 'arbitrarily big' I am trying to describe a number that in comparison with any natural number is always greater. I don't know if 'infinite' is the right word either because I've been criticised for using that word. Maths is not my native language as...
Sat Nov 26, 2016 8:41 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

This is bordering on ad hominum. It is unworthy of someone exploring mathematical puzzles and (apparent?) paradoxes. Yes, I apologise. I was getting frustrated. Thank you for responding gracefully. :oops: There is no limit to the natural numbers, but there is also no natural number that is bigger t...
Sat Nov 26, 2016 4:24 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Sure, if you define an ordered set that way. (That doesn't make ordinals not numbers though, which was my point. Perhaps one of us was just a bit sloppy using the word at one point.) And the puzzle is a way of defining an infinite set (the discards) followed by another infinite set (the keepers) in...
Sat Nov 26, 2016 11:42 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

Ordinals are numbers. ω is just not a natural number, and thus not in the original jug. Ordinals relate to the position of an element in an ordered set. They start counting from 0, so in the set {0,1,2,3,...} every element is the same as its ordinal. But if the set is ordered {1,3,5,7,0,2,4,6} the ...
Fri Nov 25, 2016 5:22 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

[given {0,2,4,6...1,3,5,7...}, you ] can never reach 1 by starting from 0 and using the successor function. If you design a single-decreasing-interval supertask that puts balls in the jug in the given order, no odd numbers will get into the jug. Do you agree? Yes, I agree. By defining this sorting ...
Fri Nov 25, 2016 10:01 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

0 2 4 6 8 ... 1 3 5 7 9 ... This is a well-ordered set of order type ω + ω. Every element has a successor (there is no largest element). Two elements lack a predecessor: 0 and 1. You can never reach 1 by starting from 0 and using the successor function. If you design a single-decreasing-interval su...
Thu Nov 24, 2016 5:25 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

The same goes for the balls in jugs. If you want to change the game to include balls and steps with numbers greater than any natural number, you can do that, but it's a different game. The game as it's defined leads to the jug being empty, because a ball with properties that allows it to remain in ...
Thu Nov 24, 2016 3:29 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 117357

Re: Infinite Balls and Jugs [solution]

If room numbers are only natural numbers and G is a room number, then G cannot exist (since that would create a contradiction) and therefore her room number does not exist and she is not in a room. This answer is consistent with the parameters of the puzzle. Your answer has to account for creating ...