Here's something I was thinking about while I was in a conversation about Hilbert's Hotel. In Bill Craig's book, Time and Eternity, he makes this point:
"Now infinite set theory is strictly logically consistent, granted its axioms and rules, but that does nothing to prove that such a system can exist in the real world. This fact is especially evident when it comes to mathematical operations such as subtraction and division, which transfinite arithmetic must prohibit in order to maintain logical consistency. While we can slap the hand of the mathematician who attempts such operations with infinite numbers, we cannot in reality prevent people from checking out of Hilbert's Hotel with all the attendent absurdities." p. 224
Let me explain one of those absurdities. Imagine in one scenario the hotel is full, and all the guests in the odd numbered rooms check out. You'd still be left with an infinite number of guests in the hotel. But now imagine that instead, all the guests in the rooms number 4, 5, 6, . . . check out. Now, you're left with three guests. In both cases, an infinite set of guests checked out, but in one case you're left with an infinite set of guests, and in the other, you're left with three guests. So, infinity minus infinity can leave you with infinity, or infinity minus infinity can leave you with three. It appears to matters which infinite set of guests checks out. Or so you'd think.
Remember the nature of countable infinities. They are all the same size. The set of all odd numbers can be put into one to one correspondence with the set of all even and odd numbers together.
So let's imagine this scenario.
Monday: Hilberts Hotel is full.
Tuesday morning: Everybody in rooms 4 and up check out. So an infinite set of peopel check out leaving three guests.
Tuesday evening: All the even numbered room from 4 up get filled, leaving all the odd numbered rooms from 5 up empty.
Wednesday morning: Everybody renews for another night.
Wednesday evening: All the people who checked out on Tuesday morning check back into the odd numbered rooms from 5 up.
Thursday morning: All the people who checked in Wednesday evening check back out, leaving rooms 1-3, and all the even numbers from 4 up occupied.
Here's the absurdity. The exact same people who checked out of a fully occupied hotel on Tuesday morning checked out of a fully occupied hotel on Thursday morning. Yet when they checked out on Tuesday, it left the hotel with only three guests, and when they checked out on Thursday, it left the hotel with an infinite set of guests.
So apparently it does not depend on which group of people checked out since it was the same group both times. What matters is which rooms they checked out of. But why should that matter?
Here's another scenario I came up with. Imagine you've got an infinite set of men and an infinite set of women, and that you pair them up in a one to one correspondence so that each couple forms a boyfriend/girlfriend relationship. Now, imagine that every other woman dies, leaving every other man single. You've still got an infinite set of couples with an infinite set of men and women. But now you've also got an infinite set of single men.
These men don't have to be alone, though. All you have to do is have all the surviving women break up with their current boyfriends. Since there are an infinite set of women, you can then put them into one to one correspondence with all the men, and now there can be one woman for every man and one man for every woman. Nobody has to be single.
The odd thing about this scenario is the fact that all that happened was that each woman broke up with one man and started going out with a different man. No woman had to go out with two men. She just had to change partners. But what difference does it make whether Lisa is dating Jack or Ned? Apparently, it makes all the differnce in the world if we're dealing with infinite sets. I don't know about you, but that strikes me as being straight up kooky dukes.
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