Our modern understanding of set cardinality came from the work of Georg Cantor in the 1890's. Sets can have three cardinalities: finite, countable and uncountable. Finite sets can be assigned a specific number as their cardinality: the number of items in the set. Both countable and uncountable sets are infinite. Cantor was the first mathematician to point out that the characteristic of an infinite set is that it can be put into a onetoone correspondence with a proper subset of itself.

Give a specific number for the cardinality of a set if it is finite. For finite sets, the cardinality is the number of items in it. For infinite sets, it is impossible to assign a specific number for the cardinality  we can only use a descriptive word. A proper subset of a set is one that contains some  but not all  of the members of the set, but nothing that is not in it. For example, a subset of the letters in the English alphabet are the letters in the word "banana." For finite sets, proper subsets are smaller than the sets. For infinite sets, this is not true.

Start at a specific element of the set and keep counting forever in a specific way to enumerate all the elements of a set. This is the definition of a countably infinite set. The key characteristic is that there is an algorithm to list all of the elements and this algorithm goes on forever. The archetypal countably infinite set is the integers. Start counting at "one" and continue with the next sequential number. You cannot give a number for the cardinality, you can only say it goes on forever. Notice that for each integer there is a corresponding even number that is twice as big. There are as many integers as there are even integers. There is a onetoone correspondence between the set and a proper subset of the set.

Compare a set to the numbers between zero and one to see if the set is uncountably infinite. You cannot start counting them, as there is no "next" number after a number between zero and one. Cantor gave an example to help with an intuitive understand uncountable sets: points and lines. Points have no length or width, yet a line is made up of points. If lines were a countable infinity of points, the length of the line would be 0 + 0 + 0 and so on forever. Lines must have an uncountable number of points.
Tips & Warnings
 Cantor's test to see if two sets have the same cardinality is whether the elements of the set could be put in a one to one correspondence with each other.
 Arithmetic only works for finite sets. If N is either countably or uncountably infinite, N + 1 = 200N = N + N = N.
References
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