Why not solve the paradox by dropping the expectation that infinitty works like finity? (And how does Cook solve the paradox?)
The book “solves” the paradox by stating that, yes, you can add an infinite number of guests to Hilbert’s hotel, even when it was full to begin with. Again, it’s only stating surprising results and if Hilbert considered it sufficiently surprising to articulate then I’m not going to argue!
It’s not that infinity doesn’t work, it’s that it struck me that it’s barren of interesting structure. Yes, infinity + infinity is still infinity. And there’s an unlimited number of infinities that are sufficiently ill-behaved that they don’t even form a set. It seems like a concept that has very little to offer.
The book “solves” the paradox by stating that, yes, you can add an infinite number of guests to Hilbert’s hotel, even when it was full to begin with. Again, it’s only stating surprising results and if Hilbert considered it sufficiently surprising to articulate then I’m not going to argue!
It’s not that infinity doesn’t work, it’s that it struck me that it’s barren of interesting structure. Yes, infinity + infinity is still infinity. And there’s an unlimited number of infinities that are sufficiently ill-behaved that they don’t even form a set. It seems like a concept that has very little to offer.