“Imagine taking all the numbers that you could reach by counting,” I said. “Then add one more, after all of them. That is infinity.”
Disclaimer: I'm an adult that doesn't understand infinity. If you "add one more" you're still counting.
Disclaimer: I'm an adult that doesn't understand infinity. If you "add one more" you're still counting.
The correct form should be "all the numbers reachable by counting in finite time".
E.g. you can count up to 100 in a minute. Up to a million in a month[1]. Up to a billion in quite a long, but finite time. You take all such numbers, and you say that a number named Omega comes just after all of them (just like million and one comes just after all numbers that are less than or equal to ine million).
Omega is your first infinte ordinal - or, simply, infinity.
And now you're counting in a new way.
- ok, take number n, it's smaller than omega, so what about n+1, n+2, etc. ?
You should be telling yourself - ok, take number n. It's smaller than omega. Now check any other number m. If this property holds for any given number, then omega is greater than any given number, and thus infinite.
The new type of counting also allows you to take a different step and say, "take a collection of things and add a new thing at the end" (a limit ordinal). It's how you get from the set of finite numbers to ω. There's nothing that's "one before" ω, but all the finite numbers are before it.
Repeat these steps as much as you want (making sure none of your sets are circularly defined), and you're counting with ordinals.