Elon Musk's favorite interview riddle

This solution is right, but counts the number of points incorrectly -- or at least not rigorously.

Infinity isn't a number, so you can't add to it, or multiply by it. So saying "one plus infinity times infinity” locations" isn't right.

First, how many "circles" are there that are 1/n miles in diameter, where n is a positive integer^1? As many as there are positive integers, so there are a countable infinity of these circles. (https://en.wikipedia.org/wiki/Countable_set)

Now, how many points are there in each circle? There's one for each real number, because a real number corresponds to a position along a line. And there are an uncountable infinity of real numbers. (https://www.maa.org/external_archive/devlin/devlin_6_01.html).

There's one other possibility, which doesn't end up applying to the earth:

If the earth was so "pointed" at the top that you could go a mile south from somewhere near (but not at) the North Pole, go a mile west, and still end up at the point you started at. This doesn't apply because by the time you get a mile south, the circle you're on is greater than a mile in circumference. So there are no such sets.

So, there are a countable infinity of sets that satisfy the problem, each set containing an uncountable infinity of points.

Plus the North Pole.

[1] Really, we're looking for how many circles are there that are one mile North of such a circle, but the earth is big enough that this doesn't change the number. If the earth was small enough that you couldn't go one mile north of some of the circles, that would matter -- for example, you can't go a mile north on an asteroid 1/2 mile in diameter.