Determine the cardinality of an infinitely large set.

How do I determine the cardinality of an infinitely large set (such as real numbers)? What is c and 2^c mean?

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... then there are just as many horses as there are men, and one says: the sets are equipotent. Thus very naturally, the concept of numbers entered our lives. We are able to notice the similarity in amount of three apples and three oranges and the distinction between three fingers and five fingers. Although we have a concept of numbers, most of us do not have a precise definition of a number. This leads us to believe that we don't need to know what a number really is, but we need to know equality and order between numbers, just as chess players are not concerned with what a knight is, but rather how it performs in the game.

Therefore, before entering into a definition of what a cardinal number is we must first introduce rules relating the size of sets as follows:

C-1: Each set A is associated with a cardinal number, denoted by card A, and for each cardinal number (η) there is a set A with card A = η.

C-2: Card A = 0 if and only if A = Φ (the empty set)

C-3: If ...