To explain to a six-year-old I would start by telling them that there are many different kinds of infinity, not just one. Some infinities are odd, others are even, and others are neither. It matters whether you are asking "how many" (cardinals) or "in what position" (ordinals). For regular finite numbers, cardinals and ordinals are (more or less) the same, but for infinities they behave differently. Then, if they want to get into the weeds, you can introduce them to transfinite ordinals, diagonalization, and all that fun stuff.
> It matters whether you are asking "how many" (cardinals) or "in what position" (ordinals)
But "even" and "odd" are all about whether you can partition something into an equal number of pairs or not. If you're asking "in what position" (ordinals), you've explicitly said you're not in the realm of counting sets of things. I would argue division makes no sense in the realm of ordinals! Everyone is saying the transfinite ordinals alternate even-odd, but those are exactly the numbers where we've stated we're only interested in position, not counting. It's not clear to me why "dividing" an ordinal number into equal pairs makes any sense. (Whereas it makes perfect sense for cardinal numbers.)
> But "even" and "odd" are all about whether you can partition something into an equal number of pairs or not.
Sez you. I can just as easily define even and odd in terms of whether or not I can arrive at a given position in a (potentially infinite) sequence taking by taking two steps at a time.
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.
