> Uncountable sets are fundamentally unphysical.
I think you didn't quite mean to say this, the real numbers being uncountable yet forming the basis for classical physics.
I think you didn't quite mean to say this, the real numbers being uncountable yet forming the basis for classical physics.
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.
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.
The set of natural numbers is countable. There are infinitely many members.
It would be better to say that infinite sets are fundamentally unphysical, since there are no actual infinities. Talk about uncountable versus countable is tangential.