Wednesday, 15 April 2015

elementary set theory - Fun quiz: where did the infinitely many candies come from?



Story 1:




Let there be a bowl A with countably infinite many of candies indexed by N. Let bowl B be empty.




  • After 1/2 unit of time, we take candy number 1 and 2 from A and put them in B. Then we eat candy 1 from bowl B.

  • After 1/4 unit of time, we take candy number 3 and 4 from A and put them in B. Then we eat candy 2 from bowl B.

  • After (1/2)n unit of time, we take candy number 2n1 and 2n from A and put them in B. Then we eat candy n from bowl B.



What happens after 1 unit of time? How many candies are there left in A, how many in B, how many have you eaten? Answer:





There are no candies in A left. For any candy corresponding to a given natural number k, one can compute the time it was eaten. Similarly there are no candies in B. We ate as many candies as the cardinality of N.




Story 2:



Let there be a bowl A with countably infinite many of candies indexed by N. Let bowl B be empty.




  • After 1/2 unit of time, we take candy number 1 and 2 from A and put them in B. Then we eat candy 1 from bowl B.


  • After 1/4 unit of time, we take candy number 3 and 4 from A and put them in B. Then we eat candy 3 from bowl B.

  • After (1/2)n unit of time, we take candy number 2n1 and 2n from A and put them in B. Then we eat candy 2n1 from bowl B.



What happens after 1 unit of time? How many candies are there left in A, how many in B, how many have you eaten? Answer:




As before, there are no candies in A left. All the candies labelled by even numbers are in B, so there are countably many. We ate all the candies labelled by odd numbers, so we ate countably many.





Question:



The main question is, in both stories we do essentially the same thing: take two candies from A to B, then eat one from B. The difference is that in the second case we have infinitely many candies left in B, but in the first case it was empty after 1 unit of time. So how did this happen?



Imagine this situation: let X be conducting the eating according to story 1 with his labeling, let Y be watching. But Y secretly has a different labeling scheme in his mind, where the candy number k in X's labeling is candy number 2k1 in Y's labeling. Then after unit time, according to Y, there should be infinitely many candies in A but according to X it should be empty. So the reality depends on the spectator?


Answer



So as you said according to person X, there will be no candies left. Lets take a look, then, at what Y observes.



From person Y's point of view:
First X moves candy 1 and 3 and from bowl A to B, then eats candy 1. Next, X moves candy 5 and 7 (3 and 4 by X's labeling) to B and eats 3. This continues for infinitely many steps. At this point person Y sees every odd candy eaten. Note that according to Y, the even candies never existed. So Y as well sees that there are no candies left in the bowls.




You may also be interested in these two links:
A strange puzzle having two possible solutions
Ross–Littlewood paradox


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