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Picking the first infinite ordinal as the infinity to explain to kids might not be the best choice for every kid, though. Oridnals are tricky. I am not too comf

Picking the first infinite ordinal as the infinity to explain to kids might not be the best choice for every kid, though. Oridnals are tricky. I am not too comfortable with them myself, and I say that as an adult with a degree in math! 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.

antidesitter 8y ago
I agree. Cardinal numbers (and bijections) are probably easier to start with.
- zhte415 8y ago
  Given Cantor made it for the battle field.. it needed to be easy to deal with.
throwawaymath 8y ago
I agree with you about Cantor's diagonalization. I think that really gives a "tactile" conception of infinity. If I had to explain infinity I'd probably choose to explain injections, surjections, and bijections; followed by countability and uncountability. If the child is old enough to understand basic addition and multiplication, you can probably run through a short explanation of the different (elementary) number systems.
- romwell 8y ago
  The cool thing is that you can introduce these concepts (bijections, etc) without calling them by their overly formal names, and yet maintaining rigor. E.g.: on a pasture, there are black and white sheep. How can you know, without counting, whether there are more black sheep than white sheep, the other way around, or there's the same number? Well, start taking them out in pairs, black and white sheep in each pair. If at some point you have a black sheep, but no white one to pair with, you know there at least as many black sheep as there are white ones. Same for the other way around. And if all sheep can come out in pairs like that, you conclude there must be the same number of them! My wording is not the clearest here, but you can get the idea. The notion of "same size" for sets via putting things side-by-side is something kids can get before they learn numbers.
gowld 8y ago
Get a sheet of graph paper. Imagine that it goes on forever in 2 of the 4 directions. Name the intersection points in order -- left to right, top to bottom. What's the name of the first point in the first row? 0 What's the name of the first point in the second row? \omega How is that more complicated or less interesting than cardinals?
- empath75 8y ago
  I guess I’m mixing up ordinals and cardinals but it seems odd that if you order the points differently ((0,0), (1,0), (0,1), (2,0), (1,1), (0,2), etc...)you never get to omega and cover all the same points.
  - cokernel 8y ago
    Yup, you never get to \omega that way. Cardinality is a coarser notion than "ordinality". So your example shows that Card(\omega * \omega) = Card(\omega) = \aleph_0, even though \omega * \omega and \omega have different order types. EDIT: Just wanted to add that an order isomorphism has two requirements: (1) it needs to be a bijection (so order-isomorphic objects have the same cardinality); and (2) it needs to preserve all inequalities (so a strict inequality among items in one object turns into a strict inequality in the same direction among the corresponding items in the other object).
- romwell 8y ago
  Well, Cantor's diagonalization argument can be used to prove the Halting theorem, which is another problem that is easy to state and is very interesting to think about. The notion of the cardinality of sets comes up everywhere in mathematics, and often enough you end up showing that something holds up to a countable number of exceptions. These two kinds of infinities - cardinality of naturals and reals - are so pervasive, you can't get away from them. But you can do a lot of math without ever having to deal with the ordinal numbers. For that reason, cardinals are more interesting to me - not just as a concept in and of itself.
Sniffnoy 8y ago
> The other kinds of infinity - cardinals, for example - are encountered early on, and there are things you can do with them. There's plenty you can do with ordinals too! Being able to interate a function transfinitely many times can be quite useful. > So, I might be biased in that, but I think that the cardinals are the most playful type of infinity. Definitely disagree. Once you know the basics, doing things with cardinals tends to be either boringly easy or impossibly hard. Ordinals, on the other hand, you can just play around with and actually get somewhere.
- romwell 8y ago
  Sure, oridnals are interesting, but in the context of explaining infinity to kids transfinite induction is probably not the easiest thing to throw into the fun basket. Even first-year undergrads often have to learn regular induction! >There's plenty you can do with ordinals too Any examples that you could introduce to a kid who just asked you "What is infinity?" - genuinely curious.
  - Sniffnoy 8y ago
    > Sure, oridnals are interesting, but in the context of explaining infinity to kids transfinite induction is probably not the easiest thing to throw into the fun basket. Even first-year undergrads often have to learn regular induction! Sure, but do you really have to explain that? When I imagine introducing kids to ordinals, like the OP talks about, I'm assuming it's basically taking the approach in, say, John Baez's blog posts on large countable ordinals[1]. [1] https://johncarlosbaez.wordpress.com/2016/06/29/large-countable-ordinals-part-1/ https://johncarlosbaez.wordpress.com/2016/06/29/large-counta... https://johncarlosbaez.wordpress.com/2016/07/04/large-countable-ordinals-part-2/ https://johncarlosbaez.wordpress.com/2016/07/04/large-counta... https://johncarlosbaez.wordpress.com/2016/07/07/large-countable-ordinals-part-3/ https://johncarlosbaez.wordpress.com/2016/07/07/large-counta... > Any examples that you could introduce to a kid who just asked you "What is infinity?" - genuinely curious. Hm, maybe not. Maybe some of the classic examples of weird things that can happen with transfinite-time processes... but explaining any of that might be hard. And also that might not really be the right time to introduce people to discontinuity. Really like I said I was basically thinking of the approach above, without application. I think it stands on its own pretty well, it's fun, you can play around with it -- it's basically the "is too, times infinity+1!" game except formalized (so it kind of comes naturally out of something kids already try to do) -- and the questions have actual answers.

antidesitter 8y ago

I agree. Cardinal numbers (and bijections) are probably easier to start with.

zhte415 8y ago
Given Cantor made it for the battle field.. it needed to be easy to deal with.

throwawaymath 8y ago

I agree with you about Cantor's diagonalization. I think that really gives a "tactile" conception of infinity. If I had to explain infinity I'd probably choose to explain injections, surjections, and bijections; followed by countability and uncountability. If the child is old enough to understand basic addition and multiplication, you can probably run through a short explanation of the different (elementary) number systems.

romwell 8y ago
The cool thing is that you can introduce these concepts (bijections, etc) without calling them by their overly formal names, and yet maintaining rigor. E.g.: on a pasture, there are black and white sheep. How can you know, without counting, whether there are more black sheep than white sheep, the other way around, or there's the same number? Well, start taking them out in pairs, black and white sheep in each pair. If at some point you have a black sheep, but no white one to pair with, you know there at least as many black sheep as there are white ones. Same for the other way around. And if all sheep can come out in pairs like that, you conclude there must be the same number of them! My wording is not the clearest here, but you can get the idea. The notion of "same size" for sets via putting things side-by-side is something kids can get before they learn numbers.

gowld 8y ago

Get a sheet of graph paper. Imagine that it goes on forever in 2 of the 4 directions. Name the intersection points in order -- left to right, top to bottom. What's the name of the first point in the first row? 0 What's the name of the first point in the second row? \omega How is that more complicated or less interesting than cardinals?

empath75 8y ago
I guess I’m mixing up ordinals and cardinals but it seems odd that if you order the points differently ((0,0), (1,0), (0,1), (2,0), (1,1), (0,2), etc...)you never get to omega and cover all the same points.
- cokernel 8y ago
  Yup, you never get to \omega that way. Cardinality is a coarser notion than "ordinality". So your example shows that Card(\omega * \omega) = Card(\omega) = \aleph_0, even though \omega * \omega and \omega have different order types. EDIT: Just wanted to add that an order isomorphism has two requirements: (1) it needs to be a bijection (so order-isomorphic objects have the same cardinality); and (2) it needs to preserve all inequalities (so a strict inequality among items in one object turns into a strict inequality in the same direction among the corresponding items in the other object).
romwell 8y ago
Well, Cantor's diagonalization argument can be used to prove the Halting theorem, which is another problem that is easy to state and is very interesting to think about. The notion of the cardinality of sets comes up everywhere in mathematics, and often enough you end up showing that something holds up to a countable number of exceptions. These two kinds of infinities - cardinality of naturals and reals - are so pervasive, you can't get away from them. But you can do a lot of math without ever having to deal with the ordinal numbers. For that reason, cardinals are more interesting to me - not just as a concept in and of itself.

Sniffnoy 8y ago

> The other kinds of infinity - cardinals, for example - are encountered early on, and there are things you can do with them. There's plenty you can do with ordinals too! Being able to interate a function transfinitely many times can be quite useful. > So, I might be biased in that, but I think that the cardinals are the most playful type of infinity. Definitely disagree. Once you know the basics, doing things with cardinals tends to be either boringly easy or impossibly hard. Ordinals, on the other hand, you can just play around with and actually get somewhere.

romwell 8y ago
Sure, oridnals are interesting, but in the context of explaining infinity to kids transfinite induction is probably not the easiest thing to throw into the fun basket. Even first-year undergrads often have to learn regular induction! >There's plenty you can do with ordinals too Any examples that you could introduce to a kid who just asked you "What is infinity?" - genuinely curious.
- Sniffnoy 8y ago
  > Sure, oridnals are interesting, but in the context of explaining infinity to kids transfinite induction is probably not the easiest thing to throw into the fun basket. Even first-year undergrads often have to learn regular induction! Sure, but do you really have to explain that? When I imagine introducing kids to ordinals, like the OP talks about, I'm assuming it's basically taking the approach in, say, John Baez's blog posts on large countable ordinals[1]. [1] https://johncarlosbaez.wordpress.com/2016/06/29/large-countable-ordinals-part-1/ https://johncarlosbaez.wordpress.com/2016/06/29/large-counta... https://johncarlosbaez.wordpress.com/2016/07/04/large-countable-ordinals-part-2/ https://johncarlosbaez.wordpress.com/2016/07/04/large-counta... https://johncarlosbaez.wordpress.com/2016/07/07/large-countable-ordinals-part-3/ https://johncarlosbaez.wordpress.com/2016/07/07/large-counta... > Any examples that you could introduce to a kid who just asked you "What is infinity?" - genuinely curious. Hm, maybe not. Maybe some of the classic examples of weird things that can happen with transfinite-time processes... but explaining any of that might be hard. And also that might not really be the right time to introduce people to discontinuity. Really like I said I was basically thinking of the approach above, without application. I think it stands on its own pretty well, it's fun, you can play around with it -- it's basically the "is too, times infinity+1!" game except formalized (so it kind of comes naturally out of something kids already try to do) -- and the questions have actual answers.