How to explain infinity to kids

The thing is, this kind of infinity just doesn't come up that often when dealing with other objects in math. Even though, as Tom Lehrer sang, one can count up to infinity - or somewhere in that vicinity - and that's mathematics, the question then is - so what?

The other kinds of infinity - cardinals, for example - are encountered early on, and there are things you can do with them.

The first time I've seen the notion of infinity was in a Russian children's book. There, they made a bijection between all the (infinite number) of points in a small segment and a larger segment - and even with all the points of an (infinite) line! I didn't really get it then; while I could find nothing wrong with the argument, it certainly looked like bullshit that a short segment could have as many points as a long one.

But some things don't have to make perfect sense right away.

The next time my understanding of infinity really improved (ignoring the notation for "x growing without bound" of calculus) was in the first year of college, with Cantor's diagonal argument. And I think that's when the picture from the book I read in kindergarten made sense, at last.

The bijection in that picture would have been boring if one could always make it. But with the diagonal argument, one sees that's not the case. That's what makes these infinities interesting and fun, to me.

So, I might be biased in that, but I think that the cardinals are the most playful type of infinity. And really the kind you can explain to kids.

One night I've had a long tea on a rooftop of a Brooklyn apartment building with a friend who is an artist, and by the sunrise, she understood Cantor's diagonal argument - and enjoyed it.

It is quite regrettable that this is the kind of knowledge that's only generally shown to math majors in college. This is reason #712889 why we need to change the way we teach and talk about mathematics.

I love ops explanation because it gives more natural picture: there are _other_ numbers beyond naturals. Then you tell kids there are also negative numbers (integers) and rational numbers and real and complex and so on. I believe that way it is much easier to grasp that very abstract (but fundamental) idea of 'number'. That it is not just 1,2,3...

Because otherwise, we get people thinking "Complex numbers do not exist, that is just a silly thing used by mathematicians that has nothing to do with the real world, it is useless."

The idea that real numbers are real, imaginary numbers are not real, and complex numbers that have a real number part and an imaginary number part are also not real is a natural consequence of unfortunate naming choices.

And I thought we are talking about kids? There are not many kids who would understand the professors way.

So isn't it first about making them understand the concept of infinity? That they can imagine it, before you bombard them with other abstract math concepts?

ω - 1 makes sense in the surreal numbers (https://en.wikipedia.org/wiki/Surreal_number). The surreal numbers are the "greatest" ordered field, in a sense. They contain all ordinal numbers, which in turn contain all cardinal numbers. The cardinality of a set is just the least ordinal it can be put into a one-to-one correspondence with (this is called the von Neumann cardinal assignment).