>>13524696If you assume the guests can be counted like that, then you're assuming the guests correspond to . Let me show you what I mean by trying to count all the numbers in with positive integers and failing miserably.
So first number is just , which is just the st number, pretty easy. Next is... well, what's next? ? But is smaller than that. And is smaller than that. And so on. There isn't really anywhere to go after that. There's actually a proof disproving counting real numbers with positive integers like this and it's called Cantor's diagonal argument. It goes something like this:
Say I have a list of every single real number that's been counted (this could be compared to the list of guests waiting to enter the park). No real number should exist outside of this list. Here are the first few digits of the first few numbers of the list.
Now let's say I wanna construct a some number which, by my initial definition, should already be in the list. I'll take the first digit of the first number and add one for my first digit of . The second digit of the second number plus one can be my second digit of and so on, where the th digit of is the th digit of the th number on the list (and if the digit I'm adding one to is , we'll just say it cycles back to ). So I've now constructed a number that differs from every number in my list at at least one digit, meaning I've created a number that isn't in my supposedly complete list. This is a self contradiction, meaning such a list cannot exist in the first place.
