Also:

I'd say that I was assuming the continuum hypothesis, except that I'm an infinite set atheist.

Not that again! Let's not mix up the map and the territory. You may not think there are any infinite sets out there in the territory, but mathematics is about the map -- or, rather, mapmaking in general. So it's a category error to jump from the conviction that the "real world" contains only finitely many things to Kroneckerian skepticism about mathematical objects.

For what it's worth, the cardinality of the set of reals is 2^aleph_0. The continuum hypothesis is precisely the statement that this is equal to aleph_1. So yeah, you're definitely assuming it.