March 3, 1845: Georg Cantor is born
Georg Cantor (1845-1918)
Scientific field: Mathematics
Known for: Set theory, Transfinite numbers
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician who founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another.
In a series of 10 papers from 1869 to 1873, Cantor dealt first with the theory of numbers; this article reflected his own fascination with the subject, his studies of Gauss, and the influence of Kronecker. On the suggestion of Heinrich Eduard Heine, a colleague at Halle who recognized his ability, Cantor then turned to the theory of trigonometric series, in which he extended the concept of real numbers. Starting from the work on trigonometric series and on the function of a complex variable done by the German mathematician Bernhard Riemann (q.v.) in 1854, Cantor in 1870 showed that such a function can be represented in only one way by a trigonometric series. Consideration of the collection of numbers (points) that would not conflict with such a representation led him, first, in 1872, to define irrational numbers in terms of convergent sequences of rational numbers (quotients of integers) and then to begin his major lifework, the theory of sets and the concept of transfinite numbers.
In 1873 Cantor demonstrated that the rational numbers, though infinite, are countable (or denumerable) because they may be placed in a one-to-one correspondence with the natural numbers (i.e., the integers, as 1, 2, 3, . . .). He showed that the set (or aggregate) of real numbers (composed of irrational and rational numbers) was infinite and uncountable. Even more paradoxically, he proved that the set of all algebraic numbers contains as many components as the set of all integers and that transcendental numbers (those that are not algebraic, as ?), which are a subset of the irrationals, are uncountable and are therefore more numerous than integers, which must be conceived as infinite.
Cantor’s theory became a whole new subject of research concerning the mathematics of the infinite (e.g., an endless series, as 1, 2, 3, . . . , and even more complicated sets), and his theory was heavily dependent on the device of the one-to-one correspondence. In thus developing new ways of asking questions concerning continuity and infinity, Cantor quickly became controversial.
In 1895–97 Cantor fully propounded his view of continuity and the infinite, including infinite ordinals and cardinals, in his best known work, “Beiträge zur Begründung der transfiniten Mengelehre” (published in English under the title “Contributions to the Founding of the Theory of Transfinite Numbers”, 1915). This work contains his conception of transfinite numbers, to which he was led by his demonstration that an infinite set may be placed in a one-to-one correspondence with one of its subsets. By the smallest transfinite cardinal number he meant the cardinal number of any set that can be placed in one-to-one correspondence with the positive integers. This transfinite number he referred to as aleph-null. Larger transfinite cardinal numbers were denoted by aleph-one, aleph-two,. . . . He then developed an arithmetic of transfinite numbers that was analogous to finite arithmetic. Thus, he further enriched the concept of infinity.(1)
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