Joel David Hamkins: preference

Transcript context

…Have you been able to sort of internalize a good intuition about countable infinity? ‘Cause that is a pretty weird thing. You can have a countably infinite set of countably infinite sets, and you can shove it all in and it still is a countable infinite set. Yeah, that’s exactly right. I mean, I guess, of course when you work with these notions that the argument of Hilbert’s Hotel becomes kind of clear, there are many, many other ways to talk about it too. For example, let’s think about, say, the integer lattice, the grid of points that you get by taking pairs of natural numbers, say, so the upper right quadrant of the integer lattice, yeah? So there’s the, you know, row zero, row one, row two and so on, column zero, column one, column two and so on, and each row and column has a countable infinity of points on it, right? So those dots, if you think about them as dots, are really the same as the train cars if you think about each column in the integer lattice, it’s a countable infinity. It’s like one train car and then there’s the next train car next to it, and then the next column next to that, the next train car. And so… But if we think about it in this grid manner, then I can imagine a- a kind of winding path winding through these grid points, like up and down the diagonals- … winding back and forth. So I start at the corner point and then I go down, up and to the left, and then down and to the right, up and to the left, down and to the right, and so on, in such a way that I’m gonna hit every grid point in- on this path. So, this gives me a way of assigning room numbers to the points. Because every grid point is going to be the Nth point on that path for some N. And that gives a correspondence between the grid points and the natural numbers themselves. So it’s a kind of different picture. I mean, before we used this 3 to the C, 5 times 5 to the S, which is a kind of, you know, overly arithmetic way to think about it. But there’s a kind of direct, you know, way to understand that it’s still a countable infinity when you have countably many countable sets because you can just start putting them on this list. And as long as you give each of the infinite collections a chance to add one more person to the list, then you’re going to accommodate everyone in any of the sets in one list. Yeah, it’s a really nice visual way to think about it. You just zigzag your way across the grid to make sure everybody’s included, that gives you kind of an algorithm for including everybody. So can you speak to the uncountable infinities?…

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