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For a child, I think the simplest kind of infinity to explain is the cardinality of integers—"how many numbers there are."

lisper 4y ago
OK, but then where do you go from there? There are infinity numbers. Then what?
- kccqzy 4y ago
  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. An old HN comment echoes the same sentiment: https://news.ycombinator.com/item?id=17677010 https://news.ycombinator.com/item?id=17677010 If we really are teaching kids, teach ordinals not cardinals.
  - lisper 4y ago
    > 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.
- Georgelemental 4y ago
  Then you explain the properties of that infinity, like how infinity + infinity = infinity, or (as per OP) that it's even.
  - lisper 4y ago
    But if infinity is even then infinity + 1 must be odd. But infinity +1 = infinity, so infinity must be odd as well as even.

lisper 4y ago

OK, but then where do you go from there? There are infinity numbers. Then what?

kccqzy 4y ago
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. An old HN comment echoes the same sentiment: https://news.ycombinator.com/item?id=17677010 https://news.ycombinator.com/item?id=17677010 If we really are teaching kids, teach ordinals not cardinals.
- lisper 4y ago
  > 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.
Georgelemental 4y ago
Then you explain the properties of that infinity, like how infinity + infinity = infinity, or (as per OP) that it's even.
- lisper 4y ago
  But if infinity is even then infinity + 1 must be odd. But infinity +1 = infinity, so infinity must be odd as well as even.