Georg Cantor (English Wikipedia)

Dauben 2004, p. 1. Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation.
Dauben 2004, pp. 8, 11, 12–13. Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation.
Dauben 2004, p. 1; Dauben 1977, p. 89 15n. Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's mental illness as "cyclic manic-depression". Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation.

adsabs.harvard.edu (Global: 14^th place; English: 9^th place)

ui.adsabs.harvard.edu

For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumerous – see Moore, A. W. (April 1995). "A brief history of infinity". Scientific American. 272 (4): 112–116 (114). Bibcode:1995SciAm.272d.112M. doi:10.1038/scientificamerican0495-112.

archive.org (Global: 6^th place; English: 6^th place)

The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9. Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837. Purkert, Walter; Ilgauds, Hans Joachim (1985). Georg Cantor: 1845–1918. Birkhäuser. ISBN 978-0-8176-1770-7.
Dauben, Joseph Warren (1979). Georg Cantor His Mathematics and Philosophy of the Infinite. Princeton University Press. pp. introduction. ISBN 9780691024479.
Dauben 1977, p. 86; Dauben 1979, pp. 120, 143. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, chpt. 6. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 280: "... the tradition made popular by Arthur Moritz Schönflies blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 248. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Reid, Constance (1996). Hilbert. New York: Springer-Verlag. p. 177. ISBN 978-0-387-04999-1.
Bruno, Leonard C.; Baker, Lawrence W. (1999). Math and mathematicians: the history of math discoveries around the world. Detroit, Mich.: U X L. p. 54. ISBN 978-0787638139. OCLC 41497065.
Dauben 1979, p. 163. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 34. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9. Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837.
Dauben 1979, p. 138. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 139. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 282. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 136; Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9. Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837.
Dauben 1979, pp. 281–283. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 283. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
For a discussion of König's paper see Dauben 1979, pp. 248–250. For Cantor's reaction, see Dauben 1979, pp. 248, 283. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, pp. 283–284. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 284. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
For this, and more information on the mathematical importance of Cantor's work on set theory, see e.g., Suppes 1972. Suppes, Patrick (1972) [1960]. Axiomatic Set Theory. New York: Dover. ISBN 978-0-486-61630-8. Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics.
Cantor (1895), Cantor (1897). The English translation is Cantor 1955. Cantor, Georg (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512. doi:10.1007/bf02124929. S2CID 177801164. Archived from the original on 23 April 2014. Cantor, Georg (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. doi:10.1007/bf01444205. S2CID 121665994. Cantor, Georg (1955) [1915]. Philip Jourdain (ed.). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover Publications.. ISBN 978-0-486-60045-1
Wallace, David Foster (2003). Everything and More: A Compact History of Infinity. New York: W. W. Norton and Company. p. 259. ISBN 978-0-393-00338-3.
Dauben 1979, pp. 69, 324 63n. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Moore and Garciadiego 1981, pp. 331, 343; Purkert 1989, p. 56. Moore, Gregory H.; Garciadiego, Alejandro (1981). "Burali-Forti's Paradox: A Reappraisal of Its Origins". Historia Mathematica. 8 (3): 319–350. doi:10.1016/0315-0860(81)90070-7. Purkert, Walter (1989). "Cantor's Views on the Foundations of Mathematics". In Rowe, David E.; McCleary, John (eds.). The History of Modern Mathematics, Volume 1. Academic Press. pp. 49–65. ISBN 978-0-12-599662-4.
Dauben 1979, p. 295. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 120. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 225 Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 266. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 296. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979, p. 144. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
"Dauben 1979, p. 96. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Grattan-Guinness 1971 (quotation from p. 350, note), Dauben 1979, p. 1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.) Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1979 Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.

arxiv.org (Global: 49^th place; English: 31^st place)

Katz, Karin Usadi; Katz, Mikhail G. (2012). "A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography". Foundations of Science. 17 (1): 51–89. arXiv:1104.0375. doi:10.1007/s10699-011-9223-1. S2CID 119250310.

asa3.org (Global: low place; English: low place)

Hedman, Bruce (1993). "Cantor's Concept of Infinity: Implications of Infinity for Contingence". Perspectives on Science and Christian Faith. 45 (1): 8–16. Retrieved 5 March 2020.

books.google.com (Global: 3^rd place; English: 3^rd place)

Suppes, Patrick (1972). Axiomatic Set Theory. Dover. p. 1. ISBN 9780486616308. With a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects.... As a consequence, many fundamental questions about the nature of mathematics may be reduced to questions about set theory.

britannica.com (Global: 37^th place; English: 64^th place)

Georg Cantor at the Encyclopedia Britannica, cited 13 August 2025.

dartmouth.edu (Global: 2,786^th place; English: 1,776^th place)

math.dartmouth.edu

Dauben, Joseph Warren (20 September 1990). "Chapter 6". GEORG CANTOR His Mathematics and Philosophy of the Infinite. Princeton University Press. ISBN 978-0691024479. Archived from the original on 5 August 2025. Retrieved 9 February 2025.

digizeitschriften.de (Global: low place; English: low place)

Hilbert (1926, p. 170): "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können." (Literally: "Out of the Paradise that Cantor created for us, no one must be able to expel us.") Hilbert, David (1926). "Über das Unendliche". Mathematische Annalen. 95: 161–190. doi:10.1007/BF01206605. S2CID 121888793.

doi.org (Global: 2^nd place; English: 2^nd place)

The biographical material in this article is mostly drawn from Dauben 1979. Grattan-Guinness 1971, and Purkert and Ilgauds 1985 are useful additional sources. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9. Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837. Purkert, Walter; Ilgauds, Hans Joachim (1985). Georg Cantor: 1845–1918. Birkhäuser. ISBN 978-0-8176-1770-7.
Dauben 1977, p. 86; Dauben 1979, pp. 120, 143. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1977, p. 102. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Dauben 2004, p. 1; Dauben 1977, p. 89 15n. Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Hilbert (1926, p. 170): "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können." (Literally: "Out of the Paradise that Cantor created for us, no one must be able to expel us.") Hilbert, David (1926). "Über das Unendliche". Mathematische Annalen. 95: 161–190. doi:10.1007/BF01206605. S2CID 121888793.
Dauben 1977, p. 89 15n. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9. Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837.
Dauben 1979, p. 136; Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9. Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837.
Johnson, Phillip E. (1972). "The Genesis and Development of Set Theory". The Two-Year College Mathematics Journal. 3 (1): 55–62. doi:10.2307/3026799. JSTOR 3026799.
Cantor 1874 Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID 199545885. Archived (PDF) from the original on 7 October 2017.
Cooke, Roger (1993). "Uniqueness of trigonometric series and descriptive set theory, 1870–1985". Archive for History of Exact Sciences. 45 (4): 281. doi:10.1007/BF01886630. S2CID 122744778.
Katz, Karin Usadi; Katz, Mikhail G. (2012). "A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography". Foundations of Science. 17 (1): 51–89. arXiv:1104.0375. doi:10.1007/s10699-011-9223-1. S2CID 119250310.
Ehrlich, P. (2006). "The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes" (PDF). Arch. Hist. Exact Sci. 60 (1): 1–121. doi:10.1007/s00407-005-0102-4. S2CID 123157068. Archived from the original (PDF) on 15 February 2013.
Cantor 1874. English translation: Ewald 1996, pp. 840–843. Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID 199545885. Archived (PDF) from the original on 7 October 2017. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.
For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumerous – see Moore, A. W. (April 1995). "A brief history of infinity". Scientific American. 272 (4): 112–116 (114). Bibcode:1995SciAm.272d.112M. doi:10.1038/scientificamerican0495-112.
For more details on Cantor's article, see Georg Cantor's first set theory article and Gray, Robert (1994). "Georg Cantor and Transcendental Numbers" (PDF). American Mathematical Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Archived from the original (PDF) on 21 January 2022. Retrieved 6 December 2013.. Gray (pp. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number.
Cantor's construction starts with the set of transcendentals T and removes a countable subset {t_n} (for example, t_n = e / n). Call this set T₀. Then T = T₀ ∪ {t_n} = T₀ ∪ {t_2n-1} ∪ {t_2n}. The set of reals R = T ∪ {a_n} = T₀ ∪ {t_n} ∪ {a_n} where a_n is the sequence of real algebraic numbers. So both T and R are the union of three pairwise disjoint sets: T₀ and two countable sets. A one-to-one correspondence between T and R is given by the function: f(t) = t if t ∈ T₀, f(t_2n-1) = t_n, and f(t_2n) = a_n. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals (Cantor 1879, p. 4). Georg Cantor (1879). "Ueber unendliche, lineare Punktmannichfaltigkeiten (1)". Mathematische Annalen. 15 (1): 1–7. doi:10.1007/bf01444101. S2CID 179177510.
Dauben 1977, p. 89. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Cantor 1883. Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen. 21 (4): 545–591. doi:10.1007/bf01446819. S2CID 121930608. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
Cantor (1895), Cantor (1897). The English translation is Cantor 1955. Cantor, Georg (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512. doi:10.1007/bf02124929. S2CID 177801164. Archived from the original on 23 April 2014. Cantor, Georg (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. doi:10.1007/bf01444205. S2CID 121665994. Cantor, Georg (1955) [1915]. Philip Jourdain (ed.). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover Publications.. ISBN 978-0-486-60045-1
Cantor 1883, pp. 587–588; English translation: Ewald 1996, pp. 916–917. Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen. 21 (4): 545–591. doi:10.1007/bf01446819. S2CID 121930608. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.
Moore 1988, pp. 52–53; Moore and Garciadiego 1981, pp. 330–331. Moore, Gregory H. (19 April 2023). "The roots of Russell's paradox". Russell: The Journal of Bertrand Russell Archives. 8 (1). Johns Hopkins University Press: 46–56. ISSN 1913-8032. Retrieved 28 March 2025. Moore, Gregory H.; Garciadiego, Alejandro (1981). "Burali-Forti's Paradox: A Reappraisal of Its Origins". Historia Mathematica. 8 (3): 319–350. doi:10.1016/0315-0860(81)90070-7.
Moore and Garciadiego 1981, pp. 331, 343; Purkert 1989, p. 56. Moore, Gregory H.; Garciadiego, Alejandro (1981). "Burali-Forti's Paradox: A Reappraisal of Its Origins". Historia Mathematica. 8 (3): 319–350. doi:10.1016/0315-0860(81)90070-7. Purkert, Walter (1989). "Cantor's Views on the Foundations of Mathematics". In Rowe, David E.; McCleary, John (eds.). The History of Modern Mathematics, Volume 1. Academic Press. pp. 49–65. ISBN 978-0-12-599662-4.
Moore 1982, pp. 158–160. Zermelo 1908, pp. 263–264; English translation: van Heijenoort 1967, p. 202. Moore, Gregory H. (1982). Zermelo's Axiom of Choice: Its Origins, Development & Influence. Springer. ISBN 978-1-4613-9480-8. Zermelo, Ernst (1908). "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/bf01449999. S2CID 120085563. van Heijenoort, Jean (1967). From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. ISBN 978-0-674-32449-7.
Zermelo 1930; English translation: Ewald 1996, pp. 1208–1233. Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF). Fundamenta Mathematicae. 16: 29–47. doi:10.4064/fm-16-1-29-47. Archived (PDF) from the original on 28 June 2004. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.
Dauben, Joseph Warren (1979). Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton University Press. doi:10.2307/j.ctv10crfh1. ISBN 9780691024479. JSTOR j.ctv10crfh1. S2CID 241372960.
Dauben, Joseph Warren (1978). "Georg Cantor: The Personal Matrix of His Mathematics". Isis. 69 (4): 548. doi:10.1086/352113. JSTOR 231091. PMID 387662. S2CID 26155985. Retrieved 5 March 2020. The religious dimension which Cantor attributed to his transfinite numbers should not be discounted as an aberration. Nor should it be forgotten or separated from his existence as a mathematician. The theological side of Cantor's set theory, though perhaps irrelevant for understanding its mathematical content, is nevertheless essential for the full understanding of his theory and why it developed in its early stages as it did.
Snapper, Ernst (1979). "The Three Crises in Mathematics: Logicism, Intuitionism and Formalism" (PDF). Mathematics Magazine. 524 (4): 207–216. doi:10.1080/0025570X.1979.11976784. Archived from the original (PDF) on 15 August 2012. Retrieved 2 April 2013.
Davenport, Anne A. (1997). "The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century". Isis. 88 (2): 263–295. doi:10.1086/383692. JSTOR 236574. S2CID 154486558.
Dauben 1977, p. 85. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Cantor 1932, p. 404. Translation in Dauben 1977, p. 95. Cantor, Georg (1932). Ernst Zermelo (ed.). "Gesammelte Abhandlungen mathematischen und philosophischen inhalts". Berlin: Springer. Archived from the original on 3 February 2014.. Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind". American Catholic Philosophical Quarterly. 83 (4): 533–553. doi:10.5840/acpq200983444.
Ferreiros, Jose (2004). "The Motives Behind Cantor's Set Theory—Physical, Biological and Philosophical Questions" (PDF). Science in Context. 17 (1–2): 49–83. doi:10.1017/S0269889704000055. hdl:11441/40318. PMID 15359485. S2CID 19040786. Archived (PDF) from the original on 21 September 2020.
Dauben 1977, pp. 91–93. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Grattan-Guinness 1971, p. 350. Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837.
Grattan-Guinness 1971 (quotation from p. 350, note), Dauben 1979, p. 1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.) Grattan-Guinness, Ivor (1971). "Towards a Biography of Georg Cantor". Annals of Science. 27 (4): 345–391. doi:10.1080/00033797100203837. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.

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Liouville, Joseph (13 May 1844). A propos de l'existence des nombres transcendants.

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Snapper, Ernst (1979). "The Three Crises in Mathematics: Logicism, Intuitionism and Formalism" (PDF). Mathematics Magazine. 524 (4): 207–216. doi:10.1080/0025570X.1979.11976784. Archived from the original (PDF) on 15 August 2012. Retrieved 2 April 2013.

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Ferreiros, Jose (2004). "The Motives Behind Cantor's Set Theory—Physical, Biological and Philosophical Questions" (PDF). Science in Context. 17 (1–2): 49–83. doi:10.1017/S0269889704000055. hdl:11441/40318. PMID 15359485. S2CID 19040786. Archived (PDF) from the original on 21 September 2020.

icm.edu.pl (Global: 1,395^th place; English: 4,083^rd place)

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Zermelo 1930; English translation: Ewald 1996, pp. 1208–1233. Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF). Fundamenta Mathematicae. 16: 29–47. doi:10.4064/fm-16-1-29-47. Archived (PDF) from the original on 28 June 2004. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.

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Moore 1988, pp. 52–53; Moore and Garciadiego 1981, pp. 330–331. Moore, Gregory H. (19 April 2023). "The roots of Russell's paradox". Russell: The Journal of Bertrand Russell Archives. 8 (1). Johns Hopkins University Press: 46–56. ISSN 1913-8032. Retrieved 28 March 2025. Moore, Gregory H.; Garciadiego, Alejandro (1981). "Burali-Forti's Paradox: A Reappraisal of Its Origins". Historia Mathematica. 8 (3): 319–350. doi:10.1016/0315-0860(81)90070-7.

jstor.org (Global: 23^rd place; English: 15^th place)

Dauben 1977, p. 86; Dauben 1979, pp. 120, 143. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842. Dauben, Joseph W. (1979). [Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. ISBN 978-0-691-02447-9.
Dauben 1977, p. 102. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Dauben 2004, p. 1; Dauben 1977, p. 89 15n. Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Dauben 1977, p. 89 15n. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Johnson, Phillip E. (1972). "The Genesis and Development of Set Theory". The Two-Year College Mathematics Journal. 3 (1): 55–62. doi:10.2307/3026799. JSTOR 3026799.
For more details on Cantor's article, see Georg Cantor's first set theory article and Gray, Robert (1994). "Georg Cantor and Transcendental Numbers" (PDF). American Mathematical Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Archived from the original (PDF) on 21 January 2022. Retrieved 6 December 2013.. Gray (pp. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number.
Dauben 1977, p. 89. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Dauben, Joseph Warren (1979). Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton University Press. doi:10.2307/j.ctv10crfh1. ISBN 9780691024479. JSTOR j.ctv10crfh1. S2CID 241372960.
Dauben, Joseph Warren (1978). "Georg Cantor: The Personal Matrix of His Mathematics". Isis. 69 (4): 548. doi:10.1086/352113. JSTOR 231091. PMID 387662. S2CID 26155985. Retrieved 5 March 2020. The religious dimension which Cantor attributed to his transfinite numbers should not be discounted as an aberration. Nor should it be forgotten or separated from his existence as a mathematician. The theological side of Cantor's set theory, though perhaps irrelevant for understanding its mathematical content, is nevertheless essential for the full understanding of his theory and why it developed in its early stages as it did.
Davenport, Anne A. (1997). "The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century". Isis. 88 (2): 263–295. doi:10.1086/383692. JSTOR 236574. S2CID 154486558.
Dauben 1977, p. 85. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Cantor 1932, p. 404. Translation in Dauben 1977, p. 95. Cantor, Georg (1932). Ernst Zermelo (ed.). "Gesammelte Abhandlungen mathematischen und philosophischen inhalts". Berlin: Springer. Archived from the original on 3 February 2014.. Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Dauben 1977, pp. 91–93. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.

maa.org (Global: 4,358^th place; English: 2,907^th place)

The Cantor Set Before Cantor Archived 29 August 2022 at the Wayback Machine Mathematical Association of America
For more details on Cantor's article, see Georg Cantor's first set theory article and Gray, Robert (1994). "Georg Cantor and Transcendental Numbers" (PDF). American Mathematical Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Archived from the original (PDF) on 21 January 2022. Retrieved 6 December 2013.. Gray (pp. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number.

nih.gov (Global: 5^th place; English: 5^th place)

pubmed.ncbi.nlm.nih.gov

Dauben, Joseph Warren (1978). "Georg Cantor: The Personal Matrix of His Mathematics". Isis. 69 (4): 548. doi:10.1086/352113. JSTOR 231091. PMID 387662. S2CID 26155985. Retrieved 5 March 2020. The religious dimension which Cantor attributed to his transfinite numbers should not be discounted as an aberration. Nor should it be forgotten or separated from his existence as a mathematician. The theological side of Cantor's set theory, though perhaps irrelevant for understanding its mathematical content, is nevertheless essential for the full understanding of his theory and why it developed in its early stages as it did.
Ferreiros, Jose (2004). "The Motives Behind Cantor's Set Theory—Physical, Biological and Philosophical Questions" (PDF). Science in Context. 17 (1–2): 49–83. doi:10.1017/S0269889704000055. hdl:11441/40318. PMID 15359485. S2CID 19040786. Archived (PDF) from the original on 21 September 2020.

ohio.edu (Global: low place; English: 6,671^st place)

Ehrlich, P. (2006). "The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes" (PDF). Arch. Hist. Exact Sci. 60 (1): 1–121. doi:10.1007/s00407-005-0102-4. S2CID 123157068. Archived from the original (PDF) on 15 February 2013.

philarchive.org (Global: low place; English: 7,457^th place)

Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind". American Catholic Philosophical Quarterly. 83 (4): 533–553. doi:10.5840/acpq200983444.

pitt.edu (Global: 1,616^th place; English: 1,079^th place)

philsci-archive.pitt.edu

Ferreiros, Jose (2004). "The Motives Behind Cantor's Set Theory—Physical, Biological and Philosophical Questions" (PDF). Science in Context. 17 (1–2): 49–83. doi:10.1017/S0269889704000055. hdl:11441/40318. PMID 15359485. S2CID 19040786. Archived (PDF) from the original on 21 September 2020.

quantamagazine.org (Global: 6,258^th place; English: 4,046^th place)

Howlett, Joseph (25 February 2026). "The Man Who Stole Infinity". Quanta Magazine. Retrieved 11 March 2026.

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api.semanticscholar.org

Hilbert (1926, p. 170): "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können." (Literally: "Out of the Paradise that Cantor created for us, no one must be able to expel us.") Hilbert, David (1926). "Über das Unendliche". Mathematische Annalen. 95: 161–190. doi:10.1007/BF01206605. S2CID 121888793.
Cantor 1874 Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID 199545885. Archived (PDF) from the original on 7 October 2017.
Cooke, Roger (1993). "Uniqueness of trigonometric series and descriptive set theory, 1870–1985". Archive for History of Exact Sciences. 45 (4): 281. doi:10.1007/BF01886630. S2CID 122744778.
Katz, Karin Usadi; Katz, Mikhail G. (2012). "A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography". Foundations of Science. 17 (1): 51–89. arXiv:1104.0375. doi:10.1007/s10699-011-9223-1. S2CID 119250310.
Ehrlich, P. (2006). "The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes" (PDF). Arch. Hist. Exact Sci. 60 (1): 1–121. doi:10.1007/s00407-005-0102-4. S2CID 123157068. Archived from the original (PDF) on 15 February 2013.
Cantor 1874. English translation: Ewald 1996, pp. 840–843. Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID 199545885. Archived (PDF) from the original on 7 October 2017. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.
Cantor's construction starts with the set of transcendentals T and removes a countable subset {t_n} (for example, t_n = e / n). Call this set T₀. Then T = T₀ ∪ {t_n} = T₀ ∪ {t_2n-1} ∪ {t_2n}. The set of reals R = T ∪ {a_n} = T₀ ∪ {t_n} ∪ {a_n} where a_n is the sequence of real algebraic numbers. So both T and R are the union of three pairwise disjoint sets: T₀ and two countable sets. A one-to-one correspondence between T and R is given by the function: f(t) = t if t ∈ T₀, f(t_2n-1) = t_n, and f(t_2n) = a_n. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals (Cantor 1879, p. 4). Georg Cantor (1879). "Ueber unendliche, lineare Punktmannichfaltigkeiten (1)". Mathematische Annalen. 15 (1): 1–7. doi:10.1007/bf01444101. S2CID 179177510.
Cantor 1883. Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen. 21 (4): 545–591. doi:10.1007/bf01446819. S2CID 121930608. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
Cantor (1895), Cantor (1897). The English translation is Cantor 1955. Cantor, Georg (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512. doi:10.1007/bf02124929. S2CID 177801164. Archived from the original on 23 April 2014. Cantor, Georg (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. doi:10.1007/bf01444205. S2CID 121665994. Cantor, Georg (1955) [1915]. Philip Jourdain (ed.). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover Publications.. ISBN 978-0-486-60045-1
Cantor 1883, pp. 587–588; English translation: Ewald 1996, pp. 916–917. Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen. 21 (4): 545–591. doi:10.1007/bf01446819. S2CID 121930608. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.
Moore 1982, pp. 158–160. Zermelo 1908, pp. 263–264; English translation: van Heijenoort 1967, p. 202. Moore, Gregory H. (1982). Zermelo's Axiom of Choice: Its Origins, Development & Influence. Springer. ISBN 978-1-4613-9480-8. Zermelo, Ernst (1908). "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/bf01449999. S2CID 120085563. van Heijenoort, Jean (1967). From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. ISBN 978-0-674-32449-7.
Dauben, Joseph Warren (1979). Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton University Press. doi:10.2307/j.ctv10crfh1. ISBN 9780691024479. JSTOR j.ctv10crfh1. S2CID 241372960.
Dauben, Joseph Warren (1978). "Georg Cantor: The Personal Matrix of His Mathematics". Isis. 69 (4): 548. doi:10.1086/352113. JSTOR 231091. PMID 387662. S2CID 26155985. Retrieved 5 March 2020. The religious dimension which Cantor attributed to his transfinite numbers should not be discounted as an aberration. Nor should it be forgotten or separated from his existence as a mathematician. The theological side of Cantor's set theory, though perhaps irrelevant for understanding its mathematical content, is nevertheless essential for the full understanding of his theory and why it developed in its early stages as it did.
Davenport, Anne A. (1997). "The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century". Isis. 88 (2): 263–295. doi:10.1086/383692. JSTOR 236574. S2CID 154486558.
Ferreiros, Jose (2004). "The Motives Behind Cantor's Set Theory—Physical, Biological and Philosophical Questions" (PDF). Science in Context. 17 (1–2): 49–83. doi:10.1017/S0269889704000055. hdl:11441/40318. PMID 15359485. S2CID 19040786. Archived (PDF) from the original on 21 September 2020.

st-and.ac.uk (Global: 4,765^th place; English: 5,091^st place)

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"Georg Cantor (1845-1918)". www-groups.dcs.st-and.ac.uk. Retrieved 14 September 2019.

st-andrews.ac.uk (Global: 1,053^rd place; English: 1,274^th place)

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"Cantor biography". www-history.mcs.st-andrews.ac.uk. Retrieved 6 October 2017.
O'Connor, John J; Robertson, Edmund F. (1998). "Georg Ferdinand Ludwig Philipp Cantor". MacTutor History of Mathematics.

mathshistory.st-andrews.ac.uk

O'Connor, JJ; Robertson, E F (October 1998). "Georg Ferdinand Ludwig Philipp Cantor". Maths History. University of St Andrews. Archived from the original on 6 February 2025. Retrieved 9 February 2025. They married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind.

stanford.edu (Global: 194^th place; English: 182^nd place)

plato.stanford.edu

Rodych 2007. Rodych, Victor (2007). "Wittgenstein's Philosophy of Mathematics". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University..

uni-goettingen.de (Global: 2,423^rd place; English: 2,569^th place)

gdz.sub.uni-goettingen.de

Cantor 1874 Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID 199545885. Archived (PDF) from the original on 7 October 2017.
Cantor 1874. English translation: Ewald 1996, pp. 840–843. Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID 199545885. Archived (PDF) from the original on 7 October 2017. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.
Cantor's construction starts with the set of transcendentals T and removes a countable subset {t_n} (for example, t_n = e / n). Call this set T₀. Then T = T₀ ∪ {t_n} = T₀ ∪ {t_2n-1} ∪ {t_2n}. The set of reals R = T ∪ {a_n} = T₀ ∪ {t_n} ∪ {a_n} where a_n is the sequence of real algebraic numbers. So both T and R are the union of three pairwise disjoint sets: T₀ and two countable sets. A one-to-one correspondence between T and R is given by the function: f(t) = t if t ∈ T₀, f(t_2n-1) = t_n, and f(t_2n) = a_n. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals (Cantor 1879, p. 4). Georg Cantor (1879). "Ueber unendliche, lineare Punktmannichfaltigkeiten (1)". Mathematische Annalen. 15 (1): 1–7. doi:10.1007/bf01444101. S2CID 179177510.
Cantor 1883. Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen. 21 (4): 545–591. doi:10.1007/bf01446819. S2CID 121930608. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
Cantor (1895), Cantor (1897). The English translation is Cantor 1955. Cantor, Georg (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512. doi:10.1007/bf02124929. S2CID 177801164. Archived from the original on 23 April 2014. Cantor, Georg (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. doi:10.1007/bf01444205. S2CID 121665994. Cantor, Georg (1955) [1915]. Philip Jourdain (ed.). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover Publications.. ISBN 978-0-486-60045-1
Cantor 1883, pp. 587–588; English translation: Ewald 1996, pp. 916–917. Georg Cantor (1883). "Ueber unendliche, lineare Punktmannichfaltigkeiten (5)". Mathematische Annalen. 21 (4): 545–591. doi:10.1007/bf01446819. S2CID 121930608. Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.
Moore 1982, pp. 158–160. Zermelo 1908, pp. 263–264; English translation: van Heijenoort 1967, p. 202. Moore, Gregory H. (1982). Zermelo's Axiom of Choice: Its Origins, Development & Influence. Springer. ISBN 978-1-4613-9480-8. Zermelo, Ernst (1908). "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/bf01449999. S2CID 120085563. van Heijenoort, Jean (1967). From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press. ISBN 978-0-674-32449-7.
Cantor 1932, p. 404. Translation in Dauben 1977, p. 95. Cantor, Georg (1932). Ernst Zermelo (ed.). "Gesammelte Abhandlungen mathematischen und philosophischen inhalts". Berlin: Springer. Archived from the original on 3 February 2014.. Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.

gdz-lucene.tc.sub.uni-goettingen.de

Cantor (1895), Cantor (1897). The English translation is Cantor 1955. Cantor, Georg (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512. doi:10.1007/bf02124929. S2CID 177801164. Archived from the original on 23 April 2014. Cantor, Georg (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. doi:10.1007/bf01444205. S2CID 121665994. Cantor, Georg (1955) [1915]. Philip Jourdain (ed.). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover Publications.. ISBN 978-0-486-60045-1

usgs.gov (Global: 119^th place; English: 145^th place)

planetarynames.wr.usgs.gov

"Cantor". Gazetteer of Planetary Nomenclature. USGS Astrogeology Research Program.

web.archive.org (Global: 1^st place; English: 1^st place)

Dauben 2004, p. 1. Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation.
Dauben 2004, pp. 8, 11, 12–13. Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation.
Dauben 2004, p. 1; Dauben 1977, p. 89 15n. Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor's mental illness as "cyclic manic-depression". Dauben, Joseph (2004) [1993]. Georg Cantor and the Battle for Transfinite Set Theory (PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22. Archived (PDF) from the original on 23 January 2018. Internet version published in Journal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article as a familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation.
O'Connor, JJ; Robertson, E F (October 1998). "Georg Ferdinand Ludwig Philipp Cantor". Maths History. University of St Andrews. Archived from the original on 6 February 2025. Retrieved 9 February 2025. They married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind.
Dauben, Joseph Warren (20 September 1990). "Chapter 6". GEORG CANTOR His Mathematics and Philosophy of the Infinite. Princeton University Press. ISBN 978-0691024479. Archived from the original on 5 August 2025. Retrieved 9 February 2025.
Cantor 1874 Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID 199545885. Archived (PDF) from the original on 7 October 2017.
The Cantor Set Before Cantor Archived 29 August 2022 at the Wayback Machine Mathematical Association of America
Ehrlich, P. (2006). "The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes" (PDF). Arch. Hist. Exact Sci. 60 (1): 1–121. doi:10.1007/s00407-005-0102-4. S2CID 123157068. Archived from the original (PDF) on 15 February 2013.
Cantor 1874. English translation: Ewald 1996, pp. 840–843. Cantor, Georg (1874). "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" (PDF). Journal für die Reine und Angewandte Mathematik. 1874 (77): 258–262. doi:10.1515/crll.1874.77.258. S2CID 199545885. Archived (PDF) from the original on 7 October 2017. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.
For more details on Cantor's article, see Georg Cantor's first set theory article and Gray, Robert (1994). "Georg Cantor and Transcendental Numbers" (PDF). American Mathematical Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Archived from the original (PDF) on 21 January 2022. Retrieved 6 December 2013.. Gray (pp. 821–822) describes a computer program that uses Cantor's constructions to generate a transcendental number.
Cantor (1895), Cantor (1897). The English translation is Cantor 1955. Cantor, Georg (1895). "Beiträge zur Begründung der transfiniten Mengenlehre (1)". Mathematische Annalen. 46 (4): 481–512. doi:10.1007/bf02124929. S2CID 177801164. Archived from the original on 23 April 2014. Cantor, Georg (1897). "Beiträge zur Begründung der transfiniten Mengenlehre (2)". Mathematische Annalen. 49 (2): 207–246. doi:10.1007/bf01444205. S2CID 121665994. Cantor, Georg (1955) [1915]. Philip Jourdain (ed.). Contributions to the Founding of the Theory of Transfinite Numbers. New York: Dover Publications.. ISBN 978-0-486-60045-1
Zermelo 1930; English translation: Ewald 1996, pp. 1208–1233. Zermelo, Ernst (1930). "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF). Fundamenta Mathematicae. 16: 29–47. doi:10.4064/fm-16-1-29-47. Archived (PDF) from the original on 28 June 2004. Ewald, William B., ed. (1996). From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press. ISBN 978-0-19-853271-2.
Snapper, Ernst (1979). "The Three Crises in Mathematics: Logicism, Intuitionism and Formalism" (PDF). Mathematics Magazine. 524 (4): 207–216. doi:10.1080/0025570X.1979.11976784. Archived from the original (PDF) on 15 August 2012. Retrieved 2 April 2013.
Cantor 1932, p. 404. Translation in Dauben 1977, p. 95. Cantor, Georg (1932). Ernst Zermelo (ed.). "Gesammelte Abhandlungen mathematischen und philosophischen inhalts". Berlin: Springer. Archived from the original on 3 February 2014.. Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix. Dauben, Joseph W. (1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite". Journal of the History of Ideas. 38 (1): 85–108. doi:10.2307/2708842. JSTOR 2708842.
Ferreiros, Jose (2004). "The Motives Behind Cantor's Set Theory—Physical, Biological and Philosophical Questions" (PDF). Science in Context. 17 (1–2): 49–83. doi:10.1017/S0269889704000055. hdl:11441/40318. PMID 15359485. S2CID 19040786. Archived (PDF) from the original on 21 September 2020.

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Bruno, Leonard C.; Baker, Lawrence W. (1999). Math and mathematicians: the history of math discoveries around the world. Detroit, Mich.: U X L. p. 54. ISBN 978-0787638139. OCLC 41497065.
Moore 1988, pp. 52–53; Moore and Garciadiego 1981, pp. 330–331. Moore, Gregory H. (19 April 2023). "The roots of Russell's paradox". Russell: The Journal of Bertrand Russell Archives. 8 (1). Johns Hopkins University Press: 46–56. ISSN 1913-8032. Retrieved 28 March 2025. Moore, Gregory H.; Garciadiego, Alejandro (1981). "Burali-Forti's Paradox: A Reappraisal of Its Origins". Historia Mathematica. 8 (3): 319–350. doi:10.1016/0315-0860(81)90070-7.

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