Finite Subsets of the Natural Numbers What are the 'finite' subsets of the natural numbers? Where do we draw the line between infinite and finite subsets?
 A: Generally, a set is finite if there is a natural number (possibly zero) that says how many elements there are in the set.
This assumes that you know how to count and what a natural number is. You're supposed to have learned that in primary school, and higher mathematics generally assumes that you already believe what you learned in primary school about counting and natural numbers. (This stands in stark contrast to several more advanced areas of mathematics where you're supposed to aggressively doubt your primary-school knowledge once you reach the university level).
There are various ways to attempt more technical definitions of counting and numbers, for use in contexts where having technical definitions is inescapably necessary. A selection of these have already been given as comments to the question. But appreciating these definitions still depends on you having an intuitive understanding of what it is you're trying to define in the first place.
(In the particular case of a set of natural numbers, we can sidestep this and say that the set is finite if and only if it is bounded, that is, if there exists some natural number that is larger than all elements in the set).
