One can define an order relation using cardinality. Cardinality does capture a notion of size of sets that corresponds to intuition for finite sets and capture

One can define an order relation using cardinality. Cardinality does capture a notion of size of sets that corresponds to intuition for finite sets and captures the idea of one infinite set being “smaller” or the same size as another. In general sets don’t have a measure on them in the sense of measure theory. Indeed, assuming the Axiom of Choice there are non measurable sets of the real numbers. But every set has a cardinality. And there is a well defined linear order of the hierarchy of cardinal numbers. I’m assuming the axiom of choice.