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This is not about normal addition. Infinity is not a real number subject to the laws of arithemtic. That's a critical point.valenumr said:But to the latter point, I would saying implying "each" to infinite sets is sketchy. It implies the rooms and guests are finitely enumerable. We can't really say infinity == infinity. What's the difference between 3*inf / inf vs. 100*inf / inf?
This is about mappings between infinite sets. And, that mappings between infinite sets are much more conceptually rich than mappings between finite sets.
For example, the set of positive even numbers is clearly a proper subset of the set of all positive integers. But, there exists a bijection (one-to-one and onto map) from one set to the other between them. In mathematical terms, two sets have the same cardinality if there exists a bijection between them. It's not the case that they must have different cardinality because one has a bijection to a proper subset of the other.
With finite sets it doesn't matter how you count. With infinite sets, you need to be more careful.
The example I gave was that if everyone is outside the hotel and they decide that the women should go in first, then it's clear that none of the men will ever get a room. As the infinite set of women will fill the hotel.
But, if we alternate between the sets of men and women, then everyone gets a room.