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HandWiki. Georg Cantor's First Set Theory Article. Encyclopedia. Available online: https://encyclopedia.pub/entry/32557 (accessed on 24 September 2026).
HandWiki. Georg Cantor's First Set Theory Article. Encyclopedia. Available at: https://encyclopedia.pub/entry/32557. Accessed September 24, 2026.
HandWiki. "Georg Cantor's First Set Theory Article" Encyclopedia, https://encyclopedia.pub/entry/32557 (accessed September 24, 2026).
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Georg Cantor's First Set Theory Article
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Georg Cantor published his first set theory article in 1874, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article also contains a proof of the existence of transcendental numbers. As early as 1930, mathematicians have disagreed on whether this proof is constructive or non-constructive. Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved. Since Cantor's proof either constructs transcendental numbers or does not, an analysis of his article can determine whether his proof is constructive or non-constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs, one that uses the uncountability of the real numbers and one that does not. Historians of mathematics have examined Cantor's article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted; he added it during proofreading. They have traced this and other facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Historians have also studied Dedekind's contributions to the article, including his contributions to the theorem on the countability of the real algebraic numbers. In addition, they have looked at the article's legacy, which includes the impact that the uncountability theorem and the concept of countability have had on mathematics.

transcendental numbers analysis infinite sets

References

  1. In letter to Dedekind dated December 25, 1873, Cantor states that he has written and submitted "a short paper" titled On a Property of the Collection of All Real Algebraic Numbers. Noether & Cavaillès 1937, p. 17; English translation: Ewald 1996, p. 847.
  2. Cantor 1874. English translation: Ewald 1996, pp. 840–843.
  3. Gray 1994, p. 828.
  4. Cantor 1874, p. 259. English translation: Ewald 1996, pp. 840–841.
  5. Cantor 1874, p. 259. English translation: Gray 1994, p. 820.
  6. Cantor 1878, p. 242.
  7. Our proof is nearly the same as the proof of Corollary 2 in Gray 1994, p. 820. The only difference is that we specify the contradiction.
  8. Cantor 1874, pp. 259–260. English translation: Ewald 1996, p. 841.
  9. Cantor 1874, pp. 260–261. English translation: Ewald 1996, pp. 841–842.
  10. Cantor 1874, p. 261. English translation: Ewald 1996, p. 842.
  11. The difference between our proof and Cantor's is that he generates the sequence of closed intervals [an, bn]. To find an + 1 and bn + 1, he must also use the open intervals (an, bn). By generating a sequence of open intervals, we avoid working with the closed intervals.
  12. Gray 1994, p. 822.
  13. This example is nearly the same as an exercise in Gray 1994, p. 823. The only difference is that this sequence contains only irreducible fractions, while Gray's sequence includes the reducible fractions of the initial interval.
  14. LeVeque 1956, pp. 154–155.
  15. LeVeque 1956, p. 174.
  16. Weisstein 2003, p. 541.
  17. Noether & Cavaillès 1937, pp. 12–13. English translation: Gray 1994, p. 827; Ewald 1996, p. 844.
  18. Noether & Cavaillès 1937, p. 18. English translation: Ewald 1996, p. 848.
  19. Noether & Cavaillès 1937, p. 13. English translation: Gray 1994, p. 827.
  20. Noether & Cavaillès 1937, pp. 14–15. English translation: Ewald 1996, pp. 845–846.
  21. Noether & Cavaillès 1937, p. 16. English translation: Gray 1994, p. 827.
  22. The beginning of our proof is derived from the proof below by restricting the numbers in this proof to the interval [a, b]. However, we derive the contradiction by using a subsequence because Cantor was using sequences in his 1873 work on countability. Satz 68. Es gibt transzendente Zahlen.Gäbe es nämlich keine transzendenten Zahlen, so wären alle Zahlen algebraisch, das Kontinuum also identisch mit der Menge aller algebraischen Zahlen. Das ist aber unmöglich, weil die Menge aller algebraischen Zahlen abzählbar ist, das Kontinuum aber nicht.[25]Translation: Theorem 68. There are transcendental numbers. If there were no transcendental numbers, then all numbers would be algebraic. Hence, the continuum would be identical to the set of all algebraic numbers. However, this is impossible because the set of all algebraic numbers is countable, but the continuum is not.
  23. Gray 1994, pp. 827–828.
  24. Perron 1921, p. 162. English translation: Gray 1994, p. 828.
  25. Fraenkel 1930, p. 237. English translation: Gray 1994, p. 823.
  26. Kaplansky 1972, p. 25.
  27. The program using the diagonal method produces [math]\displaystyle{ n }[/math] digits in [math]\displaystyle{ {\colorO}(n^2 \log^2 n \log \log n) }[/math] steps, while the program using the 1874 method requires at least [math]\displaystyle{ O(2^{\sqrt[3]}) }[/math] steps to produce [math]\displaystyle{ n }[/math] digits. (Gray 1994, pp. 822–823.)
  28. Bell 1937, pp. 568–569; Hardy & Wright 1938, p. 159 (6th ed., pp. 205–206); Birkhoff & Mac Lane 1941, p. 392, (5th ed., pp. 436–437); Spivak 1967, pp. 369–370 (4th ed., pp. 448–449).
  29. Proof is constructive: Dasgupta 2014, p. 107; Sheppard 2014, pp. 131–132. Proof is non-constructive: Jarvis 2014, p. 18; Chowdhary 2015, p. 19; Stewart 2015, p. 285; Stewart & Tall 2015, p. 333.
  30. Birkhoff & Mac Lane 1941, p. 392, (5th ed., pp. 436–437).
  31. Gray 1994, p. 828.
  32. Edwards 1989; Gray 1994, p. 828.
  33. Edwards 1989, pp. 74–75.
  34. Kronecker's opinion was: "Definitions must contain the means of reaching a decision in a finite number of steps, and existence proofs must be conducted so that the quantity in question can be calculated with any required degree of accuracy."[36] So Kronecker would accept Cantor's argument as a valid existence proof, but he would not accept its conclusion that transcendental numbers exist. For Kronecker, they do not exist because their definition contains no means for deciding in a finite number of steps whether or not a given number is transcendental.[37] To prove that Cantor's construction calculates numbers to any required degree of accuracy, we need to prove: Given a k, an n can be computed such that bn – an ≤ 1/k where (an, bn) is the n-th interval of Cantor's construction. An example of how to prove this is given in Gray 1994, p. 822. Cantor's diagonal argument provides an accuracy of 10−n after n real algebraic numbers have been calculated because each of these numbers generates one digit of the transcendental number.[38]
  35. Ferreirós 2007, p. 184.
  36. Noether & Cavaillès 1937, pp. 12–16. English translation: Ewald 1996, pp. 843–846.
  37. Dauben 1979, p. 67.
  38. Noether & Cavaillès 1937, pp. 16–17. English translation: Ewald 1996, p. 847.
  39. Grattan-Guinness 1971, p. 124.
  40. Dauben 1979, pp. 67, 308–309.
  41. See "The article" section. Also: Cantor 1874, p. 259; English translation: Ewald 1996, p. 841.
  42. Ferreirós 2007, pp. 184–185, 245.
  43. "It is unclear when his attitude changed, but there is evidence that by the mid-1880s he was accepting the conclusion that infinite sets are of different powers [cardinalities]." (Ferreirós 2007)
  44. Ferreirós 2007, p. 177.
  45. Dauben 1979, pp. 67–68.
  46. Ferreirós 2007, p. 183.
  47. Ferreirós 2007, p. 185.
  48. Ferreirós 2007, pp. 109–111, 172–174.
  49. Ferreirós 1993, p. 350.
  50. Noether & Cavaillès 1937, pp. 12–13. English translation: Ewald 1996, p. 844.
  51. Noether & Cavaillès 1937, p. 13. English translation: Ewald 1996, p. 845.
  52. Ferreirós 2007, p. 179.
  53. Noether & Cavaillès 1937, pp. 14–16, 19. English translation: Ewald 1996, pp. 845–847, 849.
  54. Ferreirós 1993, pp. 349–350; Ferreirós 2007, pp. 185–186.
  55. Cantor 1878, pp. 245–254.
  56. Cantor's method of constructing a one-to-one correspondence between the set of irrational numbers and R can be used to construct one between the set of transcendental numbers and R.[60] The construction begins with the set of transcendental numbers T and removes a countable subset {tn} (for example, tn = e/n). Let this set be T0. Then T =  T0 ∪ {tn} = T0 ∪ {t2n – 1} ∪ {t2n}, and R = T ∪ {an} = T0 ∪ {tn} ∪ {an} where an 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: g(t) = t if t ∈ T0, g(t2n – 1) = tn, and g(t2n)  = an.
  57. Ferreirós 2007, pp. 267–273.
  58. Ferreirós 2007, pp. xvi, 320–321, 324.
  59. Cantor 1878, p. 243.
  60. Hawkins 1970, pp. 103–106, 127.
  61. Hawkins 1970, pp. 118, 120–124, 127.
  62. Ferreirós 2007, pp. 362–363.
  63. Cohen 1963, pp. 1143–1144.
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