I finally finished the book “The Joy of X” and the bit on infinities was unsatisfying. specifically, the discussion on how the Hilbert Hotel has infinitely many rooms, but that is not enough rooms to assign to each of the real numbers (assuming numbers were hotel guests)
@jerry I haven't read that particular one but I have visited the Hilbert Hotel a few times!
The best way I've found to describe it is that comparing infinities is all about creating mappings, and there's an easy mapping from all rational numbers (that can be represented as a/b) into the Hotel. But an irrational number like pi can't map in -- it can't be represented as a/b (things like 22/7 are just approximations, and you never get pi exactly regardless of what large numbers you pick for the fraction).
So there's no room at the hotel for poor pi. And if you agree pi can't map in, then pi+1, pi+2, pi+3, and 2pi and 3pi, are also all irrational -- an infinite number of them -- and they also can't map in. And there's an infinite number of those little irrational bastards, so you basically have infinity-squared (aleph-1) trying to fit into infinity (aleph-0), and it just doesn't.
Which is a long setup for the joke:
Shit just got "real". :)
@crdotson it feels like the wrong way to think about it - an irrational number can’t map because it isn’t a definite number, but definitionally, the Hilbert Hotel always has one more room, and so if you ask whether there is a room for pi, the answer is yes, but it’s not the hotel’s issue that pi can’t figure out what it really is
@jerry sure, there’s room for one more. You just start the mapping at +1. The problem with pi is that it brings an infinite number of pi-derived things (pi+1, pi+2, etc.) and ALSO an infinite number of friends who do the same thing. I agree, it’s irrational by definition. :)
Add comment