One of the absurdities illustrated by Hilbert's Hotel, some say, is infinity + infinity = infinity. This is absurd in the sense that "After the infinitely many new guests check in, the number of guests will remain the same as before." I think what is meant by "the number of guests is the same as before" is that the infinities in question have the same cardinality - the same number of guests before and after.
Question: Is it true that if two infinite sets have the same cardinality, they are equal? Or is it possible that two sets have the same cardinality, yet have different numbers of members?
If it is true that infinity + infinity = some greater infinity, in what sense is the claim true that "After the infinitely many new guests check in, the number of guests will remain the same as before."
Note: The context of this question is a discussion on an objection to Hilbert's Hotel on Philosofize