## The problem with infinity.

Once again. It's irrelevant. It does not matter whether there are new rooms. There could be infinite number of new rooms in the hotel, but they are not needed to accommodate the new guest (as the paradox demonstrates). Let me try to restate the problem a little:

Hilbert's paradox of the Grand Hotel (now with 50% less surjectivity):
Imagine a hotel with countably infinite number of rooms. There's countably infinite number of guests, each with their own room. A new guest arrives and the manager is on his toes.
"Oh, my," he says, "A new guest. I can now try the Hilbert's trick. I am so lucky to be a manager of an infinite hotel." And so he takes the guest and explains to him the situation that he want's to try and accommodate the guest in the same way Hilbert did. "But can't you just give me one of the empty rooms? I'm sure there's plenty." asked the guest. "I could, but where's the fun in that?" the manager replied. And so the manager moved the guest in the lowest numbered room to the second lowest, the guest from second lowest to the third lowest and so on and so forth and while he's still moving the guests to day (it ain't easy, there's infinity of them) the new guest got himself a room. And at the end of the day only those rooms that were occupied before the new guest arrived are occupied now. And they lived happily ever after. The end.

 Nothing can reach infinity even with an infinite amount of time. Infinity isn't a number.

 Quote by nikkoo Actually, infinite means unbounded, so there would be no such edge
I don't believe this is correct. If the universe were the surface of an N-sphere with an infinite radius, it would be both infinite and bounded, no?

 Quote by audioloop there is a physical problem, in an infinite universe it have an infinite mass and hence inﬁnite inertia, no motion would be possible.
What you are saying is only true if you are talking about motion of the universe as a whole, with respect to... something other than the universe. I do not think an infinite inertia argument makes sense regarding motion within the universe.

I am also not sure an infinite universe necessarily means infinite mass, or at least if it does it is infinite of a different cardinality than the infinite volume, such that the mass density would be zero on the whole. If that we're not the case, I suspect there would be curious general relativity implications, but I admit I haven't thought this through very carefully.

 Quote by Zmunkz I don't believe this is correct. If the universe were the surface of an N-sphere with an infinite radius, it would be both infinite and bounded, no?
No. A N-sphere by definition has finite radius. A flat space has different properties, such as being unbounded.