Controversy Over Cantor's Theory - Cantor's Argument

Cantor's Argument

Cantor's 1891 argument is that there exists an infinite set (which he identifies with the set of real numbers) which has a larger number of elements, or, as he put it, has a greater 'mightiness' (Mächtigkeit), than the infinite set of finite whole numbers {1, 2, 3, ...}.

There are a number of steps in his argument, as follows:

• That the elements of no set can be put into one-to-one correspondence with all of its subsets. This is known as Cantor's theorem. It depends on very few of the assumptions of set theory, and, as John P. Mayberry puts it, is a "simple and beautiful argument" that is "pregnant with consequences". Few have seriously questioned this step of the argument.

• That the concept of "having the same number" can be captured by the idea of one-to-one correspondence. This (purely definitional) assumption is sometimes known as Hume's principle. As Frege says, "If a waiter wishes to be certain of laying exactly as many knives on a table as plates, he has no need to count either of them; all he has to do is to lay immediately to the right of every plate a knife, taking care that every knife on the table lies immediately to the right of a plate. Plates and knives are thus correlated one to one" (1884, tr. 1953, §70). Sets in such a correlation are often called equipollent, and the correlation itself is called a bijective function.

• That there exists at least one infinite set of things, usually identified with the set of all finite whole numbers or "natural numbers". This assumption (not formally specified by Cantor) is captured in formal set theory by the axiom of infinity. This assumption allows us to prove, together with Cantor's theorem, that there exists at least one set that cannot be correlated one-to-one with all its subsets. It does not prove, however, that there in fact exists any set corresponding to "all the subsets".

• That there does indeed exist a set of all subsets of the natural numbers is captured in formal set theory by the power set axiom, which says that for every set there is a set of all of its subsets. (For example, the subsets of the set {a, b} are { }, {a}, {b}, and {a, b}). This allows us to prove that there exists an infinite set which is not equipollent with the set of natural numbers. The set N of natural numbers exists (by the axiom of infinity), and so does the set R of all its subsets (by the power set axiom). By Cantor's theorem, R cannot be one-to-one correlated with N, and by Cantor's definition of number or "power", it follows that R has a different number than N. It does not prove, however, that the number of elements in R is in fact greater than the number of elements in N, for only the notion of two sets having different power has been specified; given two sets of different power, nothing so far has specified which of the two is greater.

Cantor presented a well-ordered sequence of cardinal numbers, the alephs, and attempted to prove that the power of every well-defined set ("consistent multiplicity") is an aleph; and therefore that the ordering relation among alephs determines an order among the sizes of sets. However this proof was flawed, and as Zermelo wrote, "It is precisely at this point that the weakness of the proof sketched here lies… It is precisely doubts of this kind that impelled ... proof of the well-ordering theorem purely upon the axiom of choice…"

The assumption of the axiom of choice was later shown unnecessary by the Cantor-Bernstein-Schröder theorem, which makes use of the notion of injective functions from one set to another—a correlation which associates different elements of the former set with different elements of the latter set. The theorem shows that if there is an injective function from set A to set B, and another one from B to A, then there is a bijective function from A to B, and so the sets are equipollent, by the definition we have adopted. Thus it makes sense to say that the power of one set is at least as large as another if there is an injection from the latter to the former, and this will be consistent with our definition of having the same power. Since the set of natural numbers can be embedded in its power set, but the two sets are not of the same power, as shown, we can therefore say the set of natural numbers is of lesser power than its power set. However, despite its avoidance of the axiom of choice, the proof of the Cantor-Bernstein-Schröder theorem is still not constructive, in that it does not produce a concrete bijection in general.