Density vs. Cardinality
In the thread "Of Set Theory and (False?) Claims of Logical Contradictions", socle posted:
On the other hand, there are various definitions of "density" of subsets of the natural numbers (as opposed to cardinality) that are somewhat related to your idea, so looking at finite "pieces" of sets as you were doing does have applications.Density refers to the number of elements per specified interval. That is the same as population density. Cardinality also refers to the number of elements per a specified interval, called the set. If one set has a higher density than the other then it is a given it has a higher cardinality.
BTW I don't just look at finite "pieces". Those pieces give us a pattern we can follow out for infinity and make a determination about the cardinality wrt other sets that also go to infinity.