Black Friday Sale! His mother was Roman Catholic. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. After a brief teaching assignment in a Berlin girls’ school, Cantor joined the faculty at the University of Halle, where he remained for the rest of his life, first as lecturer (paid by fees only) in 1869, then assistant professor in 1872, and full professor in 1879. Early Life And Education. He then developed an arithmetic of transfinite numbers that was analogous to finite arithmetic. While honeymooning the same year with his bride, Vally Guttman, at Interlaken, Switzerland, Cantor met Dedekind, who gave a sympathetic hearing to his new theory. Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔr/ KAN-tor; March 3 [O.S. He was a rich merchant. He died on January 6, 1918. When he argued that infinite numbers had an actual existence, he drew on ancient and medieval philosophy concerning the “actual” and “potential” infinite and also on the early religious training given him by his parents. He was born on 3 March 1845. Cantor’s father, Georg Waldemar Cantor, was a successful and cosmopolitan merchant. This led to the famous problem of the continuum hypothesis, namely, that there are no cardinal numbers between aleph-null and the cardinal number of the points on a line. By then, he was excelling in trigonometry in particular, and his mathematical brilliance was noticed by his tutors. February 19] 1845 – January 6, 1918) was a German mathematician. ( b. St. Petersburg, Russia, 3 March 1845; d. Halle, Germany, 6 January 1918), mathematics, set theory. Cantor’s parents were Danish of pure Jewish descent. Thus, he further enriched the concept of infinity. Please select which sections you would like to print: Corrections? He was awarded the title of Doctor of Mathematics in 1867. He defined the cardinal and ordinal numbers and their arithmetic. Starting from the work on trigonometric series and on the function of a complex variable done by the German mathematician Bernhard Riemann in 1854, Cantor in 1870 showed that such a function can be represented in only one way by a trigonometric series. 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. Our editors will review what you’ve submitted and determine whether to revise the article. We would like to express to you our deepest thanks for your contribution. Georg Cantor was born in 1845 in the western merchant colony of Saint Petersburg, Russia, and brought up in the city until he was eleven. He formulated set theory, which is now considered as the fundamental theory in mathematics. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm's brother) was a well-known musician and soloist in a Russian imperial orchestra. Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔr/ KAN-tor; March 3 [O.S. His mother, Marie Böhm, was from a family of musicians. His extant letters to his son attest to a cheerfulness of spirit and deep appreciation of art and religion. More Facts A famous German mathematician, Georg Cantor is known for discovering and building a hierarchy of infinite sets according to their cardinal numbers. Although mental illness, beginning about 1884, afflicted the last years of his life, Cantor remained actively at work. Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous … By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. Premium Membership is now 50% off! Cantor’s salary was low, but the estate of his father, who died in 1863, enabled him to build a house for his wife and five children. He is also known for inventing the Cantor set, which is now a fundamental theory in mathematics. Larger transfinite cardinal numbers were denoted by aleph-one, aleph-two,…. 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. She was very artistic. 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. Updates? (A set is infinite if one of its parts, or subsets, has as many objects as itself.) He showed that the set (or aggregate) of real numbers (composed of irrational and rational numbers) was infinite and uncountable. Articles from Britannica Encyclopedias for elementary and high school students. 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. Cantor's work is of great philosophical interest, a fact of which he was well aware. But when Cantor applied the device of the one-to-one correspondence (e.g., {a, b, c} to {1, 2, 3}) to study the characteristics of sets, he quickly saw that they differed in the extent of their membership, even among infinite sets. February 19] 1845 – January 6, 1918) was a German mathematician. 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 π), which are a subset of the irrationals, are uncountable and are therefore more numerous than integers, which must be conceived as infinite. Georg, the oldest of six children, was regarded as an outstanding violinist. He founded set theory and worked on transfinite numbers. This transfinite number he referred to as aleph-null. This article was most recently revised and updated by,, MacTutor History of Mathematics Archive - Biography of Georg Ferdinand Ludwig Philipp Cantor, Georg Cantor - Student Encyclopedia (Ages 11 and up), “Contributions to the Founding of the Theory of Transfinite Numbers”, “Grundlagen einer allgemeinen Mannigfaltigkeitslehre”. Partly because he had been opposed by Kronecker, he often sympathized with young, aspiring mathematicians and sought to find ways to ensure that they would not suffer as he had because of entrenched faculty members who felt threatened by new ideas. His method soon produced surprising results. Save 50% off a Britannica Premium subscription and gain access to exclusive content. 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). Georg proved an outstanding student, graduating in 1860. Both agreed that a set, whether finite or infinite, is a collection of objects (e.g., the integers, {0, ±1, ±2,…}) that share a particular property while each object retains its own individuality. Georg Cantor, in full Georg Ferdinand Ludwig Philipp Cantor, (born March 3, 1845, St. Petersburg, Russia—died January 6, 1918, Halle, Germany), German mathematician who founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another. Let us know if you have suggestions to improve this article (requires login). Cantor’s parents were Danish. In 1897 he helped to convene in Zürich the first international congress of mathematics. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". Cantor, Georg. Georg Cantor (born St Petersburg, Russia, 3 March 1845; died 6 January 1918 was an important German mathematician.


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