Differentiate finite and infinite set

Answer 1

In a finite set, you can go through the elements, count them, "1, 2, 3, ...", and eventually reach a last element. An infinite set has no last element, no matter in what order you count.

An interesting characterization of an infinite set is that an inifinite set is one that has proper subsets, that can be placed in one-to-one correspondence with the original set. For example, the set of counting numbers (1, 2, 3, 4, 5, ...) has the proper subset of the positive even numbers (2, 4, 6, 8, 10, ...). You can match the elements of the two sets one-to-one, for example:

1 <--> 2

2 <--> 4

3 <--> 6

4 <--> 8

etc.