But in reality infinity is not infinite, but it has an END. The same applies to the number of Pi. The number of Pi starts with 3,14... and so forth. One would say that this number is "Infinity", but in reality, there will always be a number in the end and a number coming after it 1+n. — Apple
If Hilbert's hotel was full then wouldn't everyone already be in a room? — Michael
Everyone indeed already is in a room both before and after they all are moved to a new room all at once. But after the move (where, e.g. everyone in room n moved to room 2*n), not every room has someone in it. All the odd-numbered room are freed. That's the apparent paradox. — Pierre-Normand
I assume this paradox only arises in the case of actual infinities? — Michael
I wonder if that would count as a reductio ad absurdum of actual infinities (despite Hilbert's defence of them)? — Michael
Luckily there are always more rooms everyone can run too. It all sounds like some sort of Hellscape mystery-adventure platformer. — TheWillowOfDarkness
If a new set of guests arrives that represents the real numbers, then, in that case, Hilbert's Hotel won't be able to accommodate them all. — Pierre-Normand
If they form an orderly queue, they can be accommodated; otherwise they will have to go to Cantor's night shelter which has infinite rooms each of infinite capacity on each of it's infinite floors. Breakfast is not provided. — unenlightened
But in reality infinity is not infinite, but it has an END. — Apple
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