True, but due to precision limits in measurements you're never actually dealing with this scenario. You always have some number of significant digits which implies you're always going to see a rational number for your measurements.
Further there are strict levels of granularity on a number of measurements, for example you run against the plancks constant when measuring a number of things. So this argument for looking at all of [0,N] falls apart if we aren't dealing with measurements that are continuous.
Further there are strict levels of granularity on a number of measurements, for example you run against the plancks constant when measuring a number of things. So this argument for looking at all of [0,N] falls apart if we aren't dealing with measurements that are continuous.