@contrex,
contrex wrote:
Did you succeed in convincing your student? Although as a young man I had never been exposed to the work of Cantor, it made perfect sense to me the first time I read about his diagonal argument.
Oh, once I started talking about "infinities" as the sizes of infinite sets he changed his mind immediately and agreed with me.
Now the problem is convincing him that sets which look like they might be different sizes (such as the set of integers, the set of even numbers, and the set of all "fractions") are in fact the same "size" (in terms of cardinality).
(His argument for why I was *right*, half a second after I told him there were multiple infinities, was that the size of {...,-1, 0, 1,2,3,...} couldn't be as large as sets got because you could always find new numbers between of your set's numbers, making the set steadily larger. But that kind of thinking would him lead to believe there were more rationals than integers, and I didn't want that.)
Pondering it aloud and reading the comments, I think the best approach might be to stick with the difference between countable and uncountable infinities. Then I can use the diagonal argument to show him the number of points in [0,1] is uncountable. It didn't make sense to me the first time I saw it, but it does now, and he'll probably understand it immediately.
I would have had a lesson plan ready if I'd known we would be talking about this at all, but this was all an aside to a discussion about probability theory. I had to explain what a
countable union of events was. (If people are wondering, this is "enrichment tutoring" not "remedial tutoring".)