Right. The problem with teaching infinity by starting with cardinal numbers is that it's either too trivial or too hard. You can establish that several other sets of numbers are identified by the same infinity but there's not much you can do.
> You can establish that several other sets of numbers are identified by the same infinity but there's not much you can do.
Well, you can introduce them to the diagonal argument and the idea of a one-to-one correspondence. That's nothing to sneeze at.
But I think the real trick here is to teach them that numbers can stand for different kinds of ideas, and in particular, they can stand for "how many" or "what position", and that these are different. I would start, not with infinity, but with negative numbers. You can't have "one less than zero" because you can't take away anything from zero. That is the definition of zero. But you can have "the thing before zero", or, to be more precise, "the thing before the zeroth thing (where the zeroth thing is the thing before the first thing)", which we call -1.
Likewise you can't have "one more than infinity" because that's just infinity. That's the definition of infinity. But you can have "the thing after infinity" (or, to be more precise, "the thing after all the things that are the nth thing for all finite values of n", which we call ω.
An old HN comment echoes the same sentiment: https://news.ycombinator.com/item?id=17677010
If we really are teaching kids, teach ordinals not cardinals.