This book is very clear that it is defining “paradox” as a surprising result, not a contradiction, and it gives a resolution for each paradox.
OK.
The chapter that left the strongest impression on me was actually the first one, on infinities. I think I’ve actually come across all of the paradoxes in that chapter — Hilbert’s hotel is pretty famous — but having them all laid out in series left me wondering whether infinity wasn’t a mistake:
The paradox in Hilbert’s hotel is that infinite quantities dont work like finite quantities … which is only surprising if you have the expectation that they should. So why is infinity itself the mistake ? Why not solve the paradox by dropping the expectation that infinitty works like finity? (And how does Cook solve the paradox?)
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.
OK.
The paradox in Hilbert’s hotel is that infinite quantities dont work like finite quantities … which is only surprising if you have the expectation that they should. So why is infinity itself the mistake ? 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.