Evans [another dog show official]: With good reason, I’m afraid. I’ve just determined that we have more dogs than people at this show.
George: That’s no puzzle, Evans. Somebody brought more than one dog.
Evans: No, George, that can’t be. We strictly enforced the rule, one dog per person max.
George: So everybody brought one dog?
Evans: Actually, George, most people are here as spectators and didn’t bring any.
George: How many people showed up?
Evans: Everybody with a ticket, and we sold a denumerable infinity of tickets.
George: How many of them brought dogs?
Evans: Anybody who was allowed to brought a dog. We worked it like a lottery. We used the ASCII character set as an alphabet and we printed a unique finite character string on each ticket, starting with all the 1-character character strings, in alphabetical order, then all the 2-character character strings, and so forth.
George: So any particular finite character string possible was printed on exactly one ticket?
Evans: Right. Now, anybody getting a ticket with a character string that happened to specify a real number was allowed to bring his dog.
George: What do you mean by “specify”.
Evans: I mean identify such that the number could be computed out to any resolution desired. For instance, the guy who got the ticket marked “pi” was allowed to bring a dog, because pi is a real number.
George: I’ve noticed that each dog has been assigned a real number. That came from the character string?
Evans: Right. We assigned each dog the number that was specified by the character string on his owner’s ticket.
George: Okay. Good system. Well, there’s got to be a denumerable infinity of dogs, then.
Evans: That’s just the problem, George. There’s a nondenumerable infinity of dogs! It’s driving me nuts!
George: Calm down, Evans. That can’t be. Look, there are a lot of dogs and people running around here. You miscounted.
Evans: But it’s not a matter of counting. Think about it, George. Any real number that can be specified by some conceivable character string is assigned to at least one dog. That’s an uncountable infinity, because if you tried to list all of these real numbers, you’d be foiled by Cantor’s antidiagonals. Any of the antidiagonals can also be specified by a character string and thus will also be assigned to a particular dog. Those dogs aren’t denumerable!
George: But the people who brought them are denumerable. And nobody brought more than one dog. I see the dilemma. That’s very confusing!
Evans: It really is. It really is… But, hey, aren’t the dogs having fun?