Here is Pour-El and Richards. Here is a more recent reference that makes my claim more explicitly. Both are gated.

What about the rest of my comment?

I'm not sure what to say. You've accused me of "confusing levels," but I'm exactly disputing the idea that sets are at a lower level than real numbers. Maybe I know how to address this:

But likewise, the standard reduction to set theory does not illuminate differential equations

...and nor does the reduction of biology to physics "illuminate" human behavior. That just isn't the point!

I don't know about human behavior, which isn't much illuminated by any subject at all. But the reduction of biology to physics absolutely does illuminate biology. Here's Feynman in six easy pieces:

Everything is made of atoms. That is the key hypothesis. The most important hypothesis in all of biology, for example, is that everything that animals do, atoms do. In other words, there is nothing that living things do that cannot be understood from the point of view that they are made of atoms acting according to the laws of physics. This was not known from the beginning: it took some experimenting and theorizing to suggest this hypothesis, but now it is accepted, and it is the most useful theory for producing new ideas in the field of biology.

You simply can't say the same thing -- even hyperbolically -- about the set-theoretic idea that everything in math is a set, made up of other sets.

Matiyasevich's book "Hilbert's 10th Problem" sketches out one way to do this.