格奥尔格·康托尔
康托尔认为，无穷可以在数学上被当作一个完成了的整体来处理：无穷集合有大小之分，在有限数之外还有可以排序和比较的超穷数。
定位
哈勒的数学家，集合论的创立者：他证明无穷集合有不同的大小，并建立了超穷数理论，弗雷格、希尔伯特、罗素和哥德尔都不得不各自对此作出回应。
核心思想
- 大小不同的无穷集合
- 超穷数
影响
影响了 3
大卫·希尔伯特
影响了
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)
依据: AI 草稿
大卫·希尔伯特 › 1862–1943
库尔特·哥德尔
影响了
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)
依据: AI 草稿
库尔特·哥德尔 › 1906–1978
批评
相互批评 1
戈特洛布·弗雷格 ⇄ 相互
批评了
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
依据: AI 草稿
受其批评
Frege’s 1892 review of Cantor’s collected papers on the transfinite (and an unpublished draft of 1890) objects to Cantor’s appeal to mental abstraction.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
依据: AI 草稿
戈特洛布·弗雷格 › 1848–1925
作品
- 1883一般集合论基础
引入良序集和超穷序数，并回应对康托尔的批评；它也作为单独的专著出版，是他的集合论成形的著作。
- 1895对超穷数理论奠基的贡献
康托尔最后的重要集合论论文，分两部分（1895年、1897年）发表于《数学年鉴》：基数、序型、良序集和序数。
立场
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.
生平与背景
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.
其他主要作品
- On a Property of the Collection of All Real Algebraic Numbers (Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, 1874)
- Foundations of a General Theory of Manifolds (Grundlagen einer allgemeinen Mannigfaltigkeitslehre, 1883)
- Contributions to the Founding of the Theory of Transfinite Numbers (Beiträge zur Begründung der transfiniten Mengenlehre, 1895/97)
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