A train with infinitely many seats, one for each rational number, stops in countably many villages, one for each positive integer, in increasing order, and then finally arrives at the city. At the first village, two women board the train. At the second village, one woman leaves the train to go visit her cousin, and two other women board the train. At the third village, one woman leaves the train to go visit her cousin, and two other women board the train. At the fourth village, and in fact at every later village, the same thing keeps happening: one woman off to visit her cousin, two new women on board the train. How many women arrive at the city?
There is not enough information to uniquely determine the answer.
Here is a situation in which $0$ women get to the city: we place the women in order according to when they boarded the train, and every time a new village is reached the woman who boarded the train farthest back in time leaves. This procedure ensures that every woman eventually leaves the train.
Here is a situation in which countably many women get to the city: every time a new village is reached one of the women who boarded at the previous village leaves. This procedure ensures that the number of women who eventually reach the city increases by $1$ at each village.
One can in fact get any finite number of women to the city.
The problem is not well posed, since you don't specify which woman leaves the train at each station. See Ross–Littlewood paradox.
This problem isn't well defined as it relies on infinity to actually exist. There are two ways of viewing this.
If we want the limit of the number of people on the train as we approach the infinitith stop, this series is +2-1+2-1..., so a number of people with the same cardinality as the integers.
The second and seemingly more correct interpretation is: "How many women stay on the train throughout the whole journey?". If we make women leave the train in the same order they board, the answer is 0, as no-one remains on the train through the whole journey. If we make the last women to board leave, then the answer is infinite.
This behaviour is not strange if you have seen the behaviour of infinite series. For example, while +2-1+2-1... approached infinity, +2-1-1+2-1-1 will always be almost 0.
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