Hacker News Reader

Comment by dnate | Hacker News Reader

> Instead we can decisively say that there is another number after infinity, which is called “infinity plus one”.

I see a problem with saying that. One of my earliest troubles when dealing with infinity in algebra was understanding, that you cannot add or subtract real numbers from infinity to make it something else.

e.g. inifnity - infinity is not 0. Suddenly saying that infinity + 1 \neq inifinity would just make it more confusing.

The article is a nice mind exercise but IMO not really helpful in explaining infinity to a child.

View on HN

“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.

You are correct. Counting here changes its meaning.

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.

[1]https://www.mathsisfun.com/activity/count-billion.html

What I hear you say is that you can count to finite numbers in finite time, and infinite numbers in infinite time. That doesn't help me in understanding infinity.

The key point here is that you shouldn't be thinking of counting as a process that happens in time. How is infinity (here, omega) defined? We just say that it comes after any other natural number. So how to think about it? You shouldn't tell yourself:

- 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.

It's two different processes. Normal counting is saying, "take a thing and add one" (a successor ordinal). It's how we get from 42 to 43. If a number can be reached from zero using just this kind of counting, we call it finite.

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.

Reply on news.ycombinator.com