The history of infinitely large quantities runs like a dark thread through the work and writings of the great mathematicians of the past 2000 years. For Greek thinkers such as Aristotle, as well as for their Hellenistic successors like Eratosthenes, the idea of an infinitely large set was incoherent or “irrational.” While all of these writers often used the notion of an “infinite” (or, more precisely, an “indefinitely long”) process, they all rejected the idea that a completed set could contain a number of items that wasn’t finite. This situation, with the idea of infinity acting as a kind of backdrop to mathematical thinking, persisted until the work of mathematician Georg Cantor (1845-1918). After Cantor’s work was based on a simple conception: Two sets are the same size if, and only if, their members can be placed into a one-to-one correspondence with each other. So the sets {Moe, Larry, Curly} and {Groucho, Chico, Harpo} are the same size because we can define a correspondence (or, in math terms, a 1-1 function) from one to the other: In cases of finite sets, this is obvious -- so obvious that we simply use “cardinal numbers” such as 3 to represent the size of all sets like the two shown here. The “cardinality” of a set is, then, just the number you reach if you count up all its members. One set is smaller than (i.e., has a smaller cardinality than) another set if you cannot find a one-to-one correspondence from it to the second set that doesn’t leave out some of that second set’s members. For example, if we add Zeppo to set B, then A is smaller than B, because Zeppo would be “leftover” from any one-to-one correspondence that included the other three Marx brothers. As soon as we start thinking about larger sets, though, things get interesting. For example, consider these two sets: A = {0, 1, 2, 3, 4, 5, … } B = {0, 10, 20, 30, 40, 50, … } On the one hand, B is clearly a subset -- and a proper subset -- of A. That is, every member of B is also a member of A, but there are many members of A that are missing from B. So, in this sense, A is larger than B. But consider the following function: Clearly, if two sets’ being the same size means that you can “pair up” their items in a one-to-one correspondence, then A and B are actually the same size -- which means they have the same cardinality. This result, that a set can be the same size as one of its parts, is not too surprising. After all, since A and B are both infinitely large, we are comfortable saying that they are the same size. What Cantor did next, though, quite literally rocked the foundations of mathematics -- especially because he did it in an area (sets) that had always been considered very elementary, even childishly simple. There is a very simple set that can be formed from any given set, called its “power set,” which consists of all subsets of the original set. For example, if A = {Moe, Larry, Curly}, then the power set of A is P(A) = {ø, {Moe}, {Larry}, {Curly}, {Moe, Larry}, {Moe, Curly}, {Larry, Curly}, {Moe, Larry, Curly}}. Notice that each set in P(A) is a subset of A and, more importantly, that P(A) is larger than A. A has a cardinality (or “size”) of 3, while P(A) has cardinality 8, which is 23. If you experiment with other sets, you’ll see that if a finite set has size n, the power set of that set will have size 2n. Now, for an infinite set A, it is obvious that the power set, P(A), will be at least as large as A itself -- for one thing, for any element x in A, there will be a corresponding set {x} which is a member of P(A). What we might expect, as with the case of A and B above, is that A and P(A) are the same size -- after all, they’re both infinitely large, right? Cantor, though proved this shocker: In fact, the cardinality of P(A) is larger than the cardinality of A! That is, he proved that any one-to-one mapping, starting from A and going to P(A) will “leave out” many, many, items of P(A) -- as if we had left out infinitely many Zeppos in the case of A and B above! The conclusion is inescapable: The two sets, A and P(A), are both infinitely large, but P(A) is strictly larger than A. Cantor referred to the “sizes” of sets like P(A) as “transfinite” numbers, and his result means that there are many of these, and that they continue to grow -- forever. Why? Well, once you have Cantor’s Theorem, which says that P(A) is always larger than A, then you can take the power set of the power set of A -- i.e., P(P(A)) --  and get a larger set. So we have a series of sets:  A is smaller than P(A), which is smaller than the P(P(A)), which is smaller than P(P(P(A))), and so on. If A itself is transfinite, then you have an unending sequence of transfinite sets, each one larger than the next. The existence of this sequence was, as noted, a shock to the mathematical world -- so much so that some great figures (Poincaré, for example) suggested that Cantor was out of his mind, or even a sinister character! For the past 100 years, however, Cantor’s work has blossomed into a vast and respected field within the boundaries of mathematical research. Further reading: Enderton, Herbert: Elements of Set Theory Halmos, Paul: Naive Set Theory Suppes, Patrick: Axiomatic Set Theory About the Author A former math teacher in Georgia, Larry Coty is now USATestprep's Math Content Team Leader. He has two daughters and resides in Tucker, GA.