Georg Cantor
Cantor held that the infinite can be treated mathematically as a completed whole: infinite sets differ in size, and beyond the finite numbers lie transfinite numbers that can be ordered and compared.
Placement
Mathematician in Halle who founded set theory: he showed that infinite sets come in different sizes and built a theory of transfinite numbers that Frege, Hilbert, Russell and Gödel each had to answer.
Key ideas
- Infinite sets of different size
- Transfinite numbers
Influence
Influenced 3
David Hilbert
influenced
Hilbert placed Cantor’s continuum problem and the well-ordering problem first in his 1900 list, and Cantor wrote to him about his views on sets in 1899. SEP “The Early Development of Set Theory” (J. Ferreirós), §§2–3 (letter to Hilbert of 15 Nov. 1899; Hilbert’s Problem 1)
Evidence: AI draft
David Hilbert › 1862–1943
Bertrand Russell
influenced ✓
Russell read Cantor intensively in his youth, and his paradox arose from a study of Cantor’s theorem. SEP “Bertrand Russell” (A. D. Irvine), on the Autobiography; SEP “The Early Development of Set Theory” (J. Ferreirós)
Evidence: also in Wikidata · Wikidata
Bertrand Russell › 1872–1970
Kurt Gödel
influenced
Gödel’s paper on Cantor’s continuum problem takes up Cantor’s open question and presses the iterative conception of sets. SEP “The Early Development of Set Theory” (J. Ferreirós), §§3–4 (Gödel on Cantor’s continuum problem, iterative conception)
Evidence: AI draft
Kurt Gödel › 1906–1978
Criticism
Mutual critique 1
Gottlob Frege ⇄ mutual
criticised
Cantor’s 1885 review of Frege’s Grundlagen objected to Frege’s attempt to define number through extensions of concepts. P. Ebert and M. Rossberg, “Cantor on Frege’s Foundations of Arithmetic”, History and Philosophy of Logic 30 (2009), 341–348
Evidence: AI draft
criticised by
Frege’s 1892 review of Cantor’s collected papers on the transfinite (and an unpublished draft of 1890) objects to Cantor’s appeal to mental abstraction. Frege, review of Cantor, Zeitschrift für Philosophie und philosophische Kritik 100 (1892), 269–272, tr. in Collected Papers (1984), pp. 178–181
Evidence: AI draft
Gottlob Frege › 1848–1925
Works
Introduces well-ordered sets and the transfinite ordinal numbers and answers Cantor’s critics; it also appeared as a separate monograph and is the work in which his set theory took its form.
Cantor’s last significant papers on set theory, in two parts (1895, 1897) in the Mathematische Annalen: cardinal numbers, order types, well-ordered sets and ordinal numbers.
Position
Cantor is on the map as the founder of set theory. Working on trigonometric series, he showed in 1874 that the real numbers cannot be put in one-to-one correspondence with the natural numbers, so that infinite sets come in different sizes. In the Foundations of a General Theory of Manifolds (1883) he introduced well-ordered sets and the transfinite ordinal numbers, and the papers of 1895 and 1897 gave the final account of cardinal and ordinal numbers. He conjectured the continuum hypothesis in 1878 and could not settle it; Hilbert put it first in his list of problems of 1900, and Gödel later wrote on it. Around 1896 he knew of the paradoxes of the ordinals and the alephs, which he told Hilbert in 1899 and did not treat as a threat. Cantor reviewed Frege’s Foundations of Arithmetic in 1885 and objected to Frege’s definition of number through extensions of concepts; Frege’s review of his collected papers in 1892 objected to his appeal to mental abstraction. Russell read Cantor intensively in his youth, and his paradox arose from a study of Cantor’s theorem.
Life & context
Georg Cantor was born on 3 March 1845 in St Petersburg and grew up in Darmstadt. He entered the Swiss Federal Polytechnic in Zurich in 1862, moved to the University of Berlin after his father’s death in 1863 and took his doctorate there in 1867 with a thesis in number theory. In 1869 he habilitated at Halle, where he spent his whole career: associate professor in 1872, full professor in 1879. Leopold Kronecker, his former teacher in Berlin, opposed his work and stood in the way of an appointment there. From 1884 Cantor suffered recurring depression with repeated stays in sanatoriums. He died on 6 January 1918 in Halle.
Further main works
- On a Property of the Collection of All Real Algebraic Numbers (Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, 1874)
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