I think he did: you don't need real numbers to formulate classical physics. Everything measurable has finite precision so you can always get away without postulating that your limits actually converge to something.
Of course, the reality is that then you would wind up with awkward limit-taking machinery in your answers. Real numbers encapsulate that complexity so you might as well use them to simplify both the notation and manipulation of limits. But you don't need to.
Yep, this is what I meant to allude to, and you've worded it much better than I could have.
Perhaps a nice way to say it is that the mathematical objects necessary for physics that I can think of are separable (such as the real numbers). Basically, whenever you have uncountable sets, they come along with some topological structure which must be handled continuously.
There's even an argument that separability is sufficient for most of mathematics let alone physics (http://arxiv.org/pdf/math/0509245.pdf) but you're going to have to work very hard to persuade me that if we're going to describe any sets as physical then uncountable ones are less physical than countable ones.
Of course, the reality is that then you would wind up with awkward limit-taking machinery in your answers. Real numbers encapsulate that complexity so you might as well use them to simplify both the notation and manipulation of limits. But you don't need to.