Viewing it as sets lacking a certain property may help explain why it's useful to know and why simply using a countable model is not preferable. Uncountability

Viewing it as sets lacking a certain property may help explain why it's useful to know and why simply using a countable model is not preferable. Uncountability means that real numbers lack certain properties. If you accept the claim of physicists that the world is best described using real numbers then this has some applications. Among the things that are impossible are things like constructing a function to pick a number for each set of real numbers. Or making an algorithm to decide two numbers are equal. Even more concretely the fact that it is incredibly hard to determine whether something is non-zero (or even nonnegative) is the bane of various numerical algorithms. Obviously you can work around these issues, but uncountability is the first sign of trouble.