Cantor Georg (1845–1918), German mathematician, one of a number of late nineteenthcentury mathematicians and philosophers (including Frege, Dedekind, Peano, Russell, and Hilbert) who transformed both mathematics and the study of its philosophical foundations. The philosophical import of Cantor’s work is threefold. First, it was primarily Cantor who turned arbitrary collections into objects of mathematical study, sets. Second, he created a coherent mathematical theory of the infinite, in particular a theory of transfinite numbers. Third, linking these, he was the first to indicate that it might be possible to present mathematics as nothing but the theory of sets, thus making set theory foundational for mathematics. This contributed to the view that the foundations of mathematics should itself become an object of mathematical study. Cantor also held to a form of principle of plenitude, the belief that all the infinities given in his theory of transfinite numbers are represented not just in mathematical (or ‘immanent’ reality), but also in the ‘transient’ reality of God’s created world.
Cantor’s main, direct achievement is his theory of transfinite numbers and infinity. He characterized (as did Frege) sameness of size in terms of one-to-one correspondence, thus accepting the paradoxical results known to Galileo and others, e.g., that the collection of all natural numbers has the same cardinality or size as that of all even numbers. He added to these surprising results by showing (1874) that there is the same number of algebraic (and thus rational) numbers as there are natural numbers, but that there are more points on a continuous line than there are natural (or rational or algebraic) numbers, thus revealing that there are at least two different kinds of infinity present in ordinary mathematics, and consequently demonstrating the need for a mathematical treatment of these infinities. This latter result is often expressed by saying that the continuum is uncountable. Cantor’s theorem of 1892 is a generalization of part of this, for it says that the set of all subsets (the power-set) of a given set must be cardinally greater than that set, thus giving rise to the possibility of indefinitely many different infinities. (The collection of all real numbers has the same size as the power-set of natural numbers.) Cantor’s theory of transfinite numbers (1880– 97) was his developed mathematical theory of infinity, with the infinite cardinal numbers (the F-, or aleph-, numbers) based on the infinite ordinal numbers that he introduced in 1880 and 1883. The F-numbers are in effect the cardinalities of infinite well-ordered sets. The theory thus generates two famous questions, whether all sets (in particular the continuum) can be well ordered, and if so which of the F-numbers represents the cardinality of the continuum. The former question was answered positively by Zermelo in 1904, though at the expense of postulating one of the most controversial principles in the history of mathematics, the axiom of choice. The latter question is the celebrated continuum problem. Cantor’s famous continuum hypothesis (CH) is his conjecture that the cardinality of the continuum is represented by F , the Cohen’s methods show that it is consistent to assume that the cardinality of the continuum is given by almost any of the vast array of F-numbers. The continuum problem is now widely considered insoluble. Cantor’s conception of set is often taken to admit the whole universe of sets as a set, thus engendering contradiction, in particular in the form of Cantor’s paradox. For Cantor’s theorem would say that the power-set of the universe must be bigger than it, while, since this powerset is a set of sets, it must be contained in the universal set, and thus can be no bigger. However, it follows from Cantor’s early (1883) considerations of what he called the ‘absolute infinite’ that none of the collections discovered later to be at the base of the paradoxes can be proper sets. Moreover, correspondence with Hilbert in 1897 and Dedekind in 1899 (see Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, 1932) shows clearly that Cantor was well aware that contradictions will arise if such collections are treated as ordinary sets. See also CONTINUUM PROBLEM , SET- THEORETIC PARADOXES , SET THEORY. M.H.