So in fact, we count them both ways, get both answers, and conclude that since each answer says that it is not the case that the one set is bigger than the other, they must be the same size?
Congratulations! I think I have, if not a perfect understanding of this, at least more of one than I had yesterday! Thanks :)
You’re welcome. I like to think that I’m good at explaining this kind of thing. ;) To give credit where credit is due, it was the long comment thread with DanArmak that helped me see what the source of your confusion was. And, indeed, all the ways of counting them matter. Mathematicians really, really hate it when you can do the same thing two different ways and get two different answers.
I learned about all this from a very interesting book I once read, which has a section on Georg Cantor, who was the one who thought up these ways of comparing the sizes of different infinite sets in the first place.
So in fact, we count them both ways, get both answers, and conclude that since each answer says that it is not the case that the one set is bigger than the other, they must be the same size?
Congratulations! I think I have, if not a perfect understanding of this, at least more of one than I had yesterday! Thanks :)
You’re welcome. I like to think that I’m good at explaining this kind of thing. ;) To give credit where credit is due, it was the long comment thread with DanArmak that helped me see what the source of your confusion was. And, indeed, all the ways of counting them matter. Mathematicians really, really hate it when you can do the same thing two different ways and get two different answers.
I learned about all this from a very interesting book I once read, which has a section on Georg Cantor, who was the one who thought up these ways of comparing the sizes of different infinite sets in the first place.