Ice cream issue in Lem's 'Extraordinary Hotel'

Question

Could you clarify the ice cream issue mentioned at the end of the story The Extraordinary Hotel (pages 189-190 here)?

Answer 1

The hotel in question is Hilbert's, not Lem's: http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel. The point is that infinite sets behave weirdly: the sets of (1) natural numbers, (2) even numbers, and (3) natural numbers greater than 17 (for example) all have the same cardinality (size).

More generally: many things you might do to an infinite set - adding a finite set, or doubling it - will not change its size. The ice cream issue is an instance of this latter fact: since there are infinitely many guests, "twice as much ice cream" is the same amount of ice cream!

(NOTE: this is not provable without a certain amount of the axiom of choice! But it is true regardless of choice for all "natural" infinite sets. In particular, doubling the set of natural numbers doesn't change its size, and this is the specific case Lem and Hilbert describe.)

Answer 2

Each guest was hungry and wanted two portions, so after the ice cream was served, guest $n$ moved to chair $2 \cdot n$ and ate both his own ice cream and the ice cream at the place to his right.