Von Neumann–Bernays–Gödel set theory (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Von Neumann–Bernays–Gödel set theory" in English language version.

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Bourbaki 2004, p. 71. Bourbaki, Nicolas (2004), Elements of Mathematics: Theory of Sets, Springer, ISBN 978-3-540-22525-6.

books.google.com

von Neumann 1923. Von Neumann's definition also used the theory of well-ordered sets. Later, his definition was simplified to the current one: An ordinal is a transitive set that is well-ordered by ∈.^[44] von Neumann, John (1923), "Zur Einführung der transfiniten Zahlen", Acta Litt. Acad. Sc. Szeged X., 1: 199–208.
- Gödel's consistency proof builds the constructible universe. To build this in ZF requires some model theory. Gödel built it in NBG without model theory. For Gödel's construction, see Gödel 1940, pp. 35–46 or Cohen 1966, pp. 99–103. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).

britannica.com

"The Neumann-Bernays-Gödel axioms". Encyclopædia Britannica. Retrieved 17 January 2019.

bu.edu

math.bu.edu

For $V$ having a well-ordering implying global choice, see Implications of the axiom of limitation of size. For global choice implying the well-ordering of any class, see Kanamori 2009, p. 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
Both conclusions follow from the conclusion that every proper class can be put into one-to-one correspondence with the class of all ordinals. A proof of this is outlined in Kanamori 2009, p. 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
Kanamori 2009, p. 56; Bernays 1937, p. 69; Gödel 1940, pp. 5, 9; Mendelson 1997, p. 231. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
- Gödel 1940, p. 6; Kanamori 2012, p. 70. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
  - Kanamori 2009, p. 57; Gödel 2003, p. 121. Both references contain Gödel's proof but Kanamori's is easier to follow since he uses modern terminology. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (2003), Collected Works, Volume 4: Correspondence A–G, Oxford University Press, ISBN 978-0-19-850073-5.
  - Kanamori 2009, p. 48; Gödel 2003, pp. 104–115. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (2003), Collected Works, Volume 4: Correspondence A–G, Oxford University Press, ISBN 978-0-19-850073-5.
  - Kanamori 2012, p. 62. Kanamori, Akihiro (2012), "In Praise of Replacement" (PDF), Bulletin of Symbolic Logic, 18 (1): 46–90, doi:10.2178/bsl/1327328439, JSTOR 41472440, S2CID 18951854.
  - Kanamori 2012, p. 61 Kanamori, Akihiro (2012), "In Praise of Replacement" (PDF), Bulletin of Symbolic Logic, 18 (1): 46–90, doi:10.2178/bsl/1327328439, JSTOR 41472440, S2CID 18951854.
  - Kanamori 2009, pp. 53–54. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
  - Kanamori 2009, pp. 49, 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
  - Kanamori 2009, pp. 48, 58. Bernays' articles are reprinted in Müller 1976, pp. 1–117. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Müller, Gurt, ed. (1976), Sets and Classes: On the Work of Paul Bernays, Studies in Logic and the Foundations of Mathematics Volume 84, Amsterdam: North Holland, ISBN 978-0-7204-2284-9.
  - Kanamori 2009, pp. 48–54. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
  - Kanamori 2009, p. 56. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
  - Kanamori 2009, pp. 56–58; Gödel 1940, Chapter I: The axioms of abstract set theory, pp. 3–7. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
    - Kanamori 2009, p. 57. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, p. 65: "Forcing itself went a considerable distance in downgrading any formal theory of classes because of the added encumbrance of having to specify the classes of generic extensions." Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.

doi.org

For $V$ having a well-ordering implying global choice, see Implications of the axiom of limitation of size. For global choice implying the well-ordering of any class, see Kanamori 2009, p. 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
Both conclusions follow from the conclusion that every proper class can be put into one-to-one correspondence with the class of all ordinals. A proof of this is outlined in Kanamori 2009, p. 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
Bernays 1937, pp. 66–67. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862.
Bernays 1937, p. 66. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862.
Kanamori 2009, p. 56; Bernays 1937, p. 69; Gödel 1940, pp. 5, 9; Mendelson 1997, p. 231. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
- Bernays' axiom b(3) (Bernays 1937, p. 5). Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862.
- Gödel 1940, p. 6; Kanamori 2012, p. 70. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
  - Kanamori 2009, p. 57; Gödel 2003, p. 121. Both references contain Gödel's proof but Kanamori's is easier to follow since he uses modern terminology. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (2003), Collected Works, Volume 4: Correspondence A–G, Oxford University Press, ISBN 978-0-19-850073-5.
  - Gray 1991. Gray, Robert (1991), "Computer programs and mathematical proofs", The Mathematical Intelligencer, 13 (4): 45–48, doi:10.1007/BF03028342, S2CID 121229549.
  - Bernays 1941, p. 2; Gödel 1940, p. 5). Bernays, Paul (1941), "A System of Axiomatic Set Theory—Part II", The Journal of Symbolic Logic, 6 (1): 1–17, doi:10.2307/2267281, JSTOR 2267281, S2CID 250344277. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
    - Kanamori 2009, p. 48; Gödel 2003, pp. 104–115. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (2003), Collected Works, Volume 4: Correspondence A–G, Oxford University Press, ISBN 978-0-19-850073-5.
    - von Neumann 1925, von Neumann 1928. von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240. von Neumann, John (1928), "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift, 27: 669–752, doi:10.1007/bf01171122, S2CID 123492324.
    - Kanamori 2012, p. 62. Kanamori, Akihiro (2012), "In Praise of Replacement" (PDF), Bulletin of Symbolic Logic, 18 (1): 46–90, doi:10.2178/bsl/1327328439, JSTOR 41472440, S2CID 18951854.
    - Kanamori 2012, p. 61 Kanamori, Akihiro (2012), "In Praise of Replacement" (PDF), Bulletin of Symbolic Logic, 18 (1): 46–90, doi:10.2178/bsl/1327328439, JSTOR 41472440, S2CID 18951854.
    - Kanamori 2009, pp. 53–54. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Bernays 1941, p. 6. Bernays, Paul (1941), "A System of Axiomatic Set Theory—Part II", The Journal of Symbolic Logic, 6 (1): 1–17, doi:10.2307/2267281, JSTOR 2267281, S2CID 250344277.
    - Kanamori 2009, pp. 49, 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, pp. 48, 58. Bernays' articles are reprinted in Müller 1976, pp. 1–117. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Müller, Gurt, ed. (1976), Sets and Classes: On the Work of Paul Bernays, Studies in Logic and the Foundations of Mathematics Volume 84, Amsterdam: North Holland, ISBN 978-0-7204-2284-9.
    - Bernays 1937, p. 65. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862.
    - Kanamori 2009, pp. 48–54. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, p. 56. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, pp. 56–58; Gödel 1940, Chapter I: The axioms of abstract set theory, pp. 3–7. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
      - Kanamori 2009, p. 57. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
      - Cohen 1963. Cohen, Paul (1963), "The Independence of the Continuum Hypothesis", Proceedings of the National Academy of Sciences of the United States of America, 50 (6): 1143–1148, Bibcode:1963PNAS...50.1143C, doi:10.1073/pnas.50.6.1143, PMC 221287, PMID 16578557.
      - Kanamori 2009, p. 65: "Forcing itself went a considerable distance in downgrading any formal theory of classes because of the added encumbrance of having to specify the classes of generic extensions." Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
      - Ferreirós 2007, pp. 381–382; Cohen 1966, p. 77; Felgner 1971. Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought (2nd revised ed.), Basel, Switzerland: Birkhäuser, ISBN 978-3-7643-8349-7. Cohen, Paul (1966), Set Theory and the Continuum Hypothesis, W. A. Benjamin.
        Cohen, Paul (2008), Set Theory and the Continuum Hypothesis, Dover Publications, ISBN 978-0-486-46921-8. Felgner, Ulrich (1971), "Comparison of the axioms of local and universal choice" (PDF), Fundamenta Mathematicae, 71: 43–62, doi:10.4064/fm-71-1-43-62.
        Mostowski 1950, p. 113, footnote 11. Footnote references Wang's NQ set theory, which later evolved into MK. Mostowski, Andrzej (1950), "Some impredicative definitions in the axiomatic set theory" (PDF), Fundamenta Mathematicae, 37: 111–124, doi:10.4064/fm-37-1-111-124.

harvard.edu

ui.adsabs.harvard.edu

Cohen 1963. Cohen, Paul (1963), "The Independence of the Continuum Hypothesis", Proceedings of the National Academy of Sciences of the United States of America, 50 (6): 1143–1148, Bibcode:1963PNAS...50.1143C, doi:10.1073/pnas.50.6.1143, PMC 221287, PMID 16578557.

icm.edu.pl

matwbn.icm.edu.pl

Ferreirós 2007, pp. 381–382; Cohen 1966, p. 77; Felgner 1971. Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought (2nd revised ed.), Basel, Switzerland: Birkhäuser, ISBN 978-3-7643-8349-7. Cohen, Paul (1966), Set Theory and the Continuum Hypothesis, W. A. Benjamin.
- Mostowski 1950, p. 113, footnote 11. Footnote references Wang's NQ set theory, which later evolved into MK. Mostowski, Andrzej (1950), "Some impredicative definitions in the axiomatic set theory" (PDF), Fundamenta Mathematicae, 37: 111–124, doi:10.4064/fm-37-1-111-124.

jstor.org

For $V$ having a well-ordering implying global choice, see Implications of the axiom of limitation of size. For global choice implying the well-ordering of any class, see Kanamori 2009, p. 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
Both conclusions follow from the conclusion that every proper class can be put into one-to-one correspondence with the class of all ordinals. A proof of this is outlined in Kanamori 2009, p. 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
Bernays 1937, pp. 66–67. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862.
Bernays 1937, p. 66. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862.
Kanamori 2009, p. 56; Bernays 1937, p. 69; Gödel 1940, pp. 5, 9; Mendelson 1997, p. 231. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
- Bernays' axiom b(3) (Bernays 1937, p. 5). Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862.
- Gödel 1940, p. 6; Kanamori 2012, p. 70. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
  - Kanamori 2009, p. 57; Gödel 2003, p. 121. Both references contain Gödel's proof but Kanamori's is easier to follow since he uses modern terminology. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (2003), Collected Works, Volume 4: Correspondence A–G, Oxford University Press, ISBN 978-0-19-850073-5.
  - Bernays 1941, p. 2; Gödel 1940, p. 5). Bernays, Paul (1941), "A System of Axiomatic Set Theory—Part II", The Journal of Symbolic Logic, 6 (1): 1–17, doi:10.2307/2267281, JSTOR 2267281, S2CID 250344277. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
    - Kanamori 2009, p. 48; Gödel 2003, pp. 104–115. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (2003), Collected Works, Volume 4: Correspondence A–G, Oxford University Press, ISBN 978-0-19-850073-5.
    - Kanamori 2012, p. 62. Kanamori, Akihiro (2012), "In Praise of Replacement" (PDF), Bulletin of Symbolic Logic, 18 (1): 46–90, doi:10.2178/bsl/1327328439, JSTOR 41472440, S2CID 18951854.
    - Kanamori 2012, p. 61 Kanamori, Akihiro (2012), "In Praise of Replacement" (PDF), Bulletin of Symbolic Logic, 18 (1): 46–90, doi:10.2178/bsl/1327328439, JSTOR 41472440, S2CID 18951854.
    - Kanamori 2009, pp. 53–54. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Bernays 1941, p. 6. Bernays, Paul (1941), "A System of Axiomatic Set Theory—Part II", The Journal of Symbolic Logic, 6 (1): 1–17, doi:10.2307/2267281, JSTOR 2267281, S2CID 250344277.
    - Kanamori 2009, pp. 49, 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, pp. 48, 58. Bernays' articles are reprinted in Müller 1976, pp. 1–117. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Müller, Gurt, ed. (1976), Sets and Classes: On the Work of Paul Bernays, Studies in Logic and the Foundations of Mathematics Volume 84, Amsterdam: North Holland, ISBN 978-0-7204-2284-9.
    - Bernays 1937, p. 65. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862.
    - Kanamori 2009, pp. 48–54. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, p. 56. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, pp. 56–58; Gödel 1940, Chapter I: The axioms of abstract set theory, pp. 3–7. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
      - Kanamori 2009, p. 57. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
      - Kanamori 2009, p. 65: "Forcing itself went a considerable distance in downgrading any formal theory of classes because of the added encumbrance of having to specify the classes of generic extensions." Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.

narod.ru

bbi-math.narod.ru

von Neumann 1923. Von Neumann's definition also used the theory of well-ordered sets. Later, his definition was simplified to the current one: An ordinal is a transitive set that is well-ordered by ∈.^[44] von Neumann, John (1923), "Zur Einführung der transfiniten Zahlen", Acta Litt. Acad. Sc. Szeged X., 1: 199–208.

nih.gov

ncbi.nlm.nih.gov

Cohen 1963. Cohen, Paul (1963), "The Independence of the Continuum Hypothesis", Proceedings of the National Academy of Sciences of the United States of America, 50 (6): 1143–1148, Bibcode:1963PNAS...50.1143C, doi:10.1073/pnas.50.6.1143, PMC 221287, PMID 16578557.

pubmed.ncbi.nlm.nih.gov

Cohen 1963. Cohen, Paul (1963), "The Independence of the Continuum Hypothesis", Proceedings of the National Academy of Sciences of the United States of America, 50 (6): 1143–1148, Bibcode:1963PNAS...50.1143C, doi:10.1073/pnas.50.6.1143, PMC 221287, PMID 16578557.

sciencedirect.com

A. Enayat, "Set theoretical analogues of the Barwise-Schlipf theorem". Annals of Pure and Applied Logic vol. 173 (2022).

semanticscholar.org

api.semanticscholar.org

For $V$ having a well-ordering implying global choice, see Implications of the axiom of limitation of size. For global choice implying the well-ordering of any class, see Kanamori 2009, p. 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
Both conclusions follow from the conclusion that every proper class can be put into one-to-one correspondence with the class of all ordinals. A proof of this is outlined in Kanamori 2009, p. 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
Kanamori 2009, p. 56; Bernays 1937, p. 69; Gödel 1940, pp. 5, 9; Mendelson 1997, p. 231. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Bernays, Paul (1937), "A System of Axiomatic Set Theory—Part I", The Journal of Symbolic Logic, 2 (1): 65–77, doi:10.2307/2268862, JSTOR 2268862. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
- Gödel 1940, p. 6; Kanamori 2012, p. 70. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
  - Kanamori 2009, p. 57; Gödel 2003, p. 121. Both references contain Gödel's proof but Kanamori's is easier to follow since he uses modern terminology. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (2003), Collected Works, Volume 4: Correspondence A–G, Oxford University Press, ISBN 978-0-19-850073-5.
  - Gray 1991. Gray, Robert (1991), "Computer programs and mathematical proofs", The Mathematical Intelligencer, 13 (4): 45–48, doi:10.1007/BF03028342, S2CID 121229549.
  - Bernays 1941, p. 2; Gödel 1940, p. 5). Bernays, Paul (1941), "A System of Axiomatic Set Theory—Part II", The Journal of Symbolic Logic, 6 (1): 1–17, doi:10.2307/2267281, JSTOR 2267281, S2CID 250344277. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
    - Kanamori 2009, p. 48; Gödel 2003, pp. 104–115. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (2003), Collected Works, Volume 4: Correspondence A–G, Oxford University Press, ISBN 978-0-19-850073-5.
    - von Neumann 1925, von Neumann 1928. von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240. von Neumann, John (1928), "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift, 27: 669–752, doi:10.1007/bf01171122, S2CID 123492324.
    - Kanamori 2012, p. 62. Kanamori, Akihiro (2012), "In Praise of Replacement" (PDF), Bulletin of Symbolic Logic, 18 (1): 46–90, doi:10.2178/bsl/1327328439, JSTOR 41472440, S2CID 18951854.
    - Kanamori 2012, p. 61 Kanamori, Akihiro (2012), "In Praise of Replacement" (PDF), Bulletin of Symbolic Logic, 18 (1): 46–90, doi:10.2178/bsl/1327328439, JSTOR 41472440, S2CID 18951854.
    - Kanamori 2009, pp. 53–54. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Bernays 1941, p. 6. Bernays, Paul (1941), "A System of Axiomatic Set Theory—Part II", The Journal of Symbolic Logic, 6 (1): 1–17, doi:10.2307/2267281, JSTOR 2267281, S2CID 250344277.
    - Kanamori 2009, pp. 49, 53. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, pp. 48, 58. Bernays' articles are reprinted in Müller 1976, pp. 1–117. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Müller, Gurt, ed. (1976), Sets and Classes: On the Work of Paul Bernays, Studies in Logic and the Foundations of Mathematics Volume 84, Amsterdam: North Holland, ISBN 978-0-7204-2284-9.
    - Kanamori 2009, pp. 48–54. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, p. 56. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
    - Kanamori 2009, pp. 56–58; Gödel 1940, Chapter I: The axioms of abstract set theory, pp. 3–7. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244. Gödel, Kurt (1940), The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory (Revised ed.), Princeton University Press, ISBN 978-0-691-07927-1 {{citation}}: ISBN / Date incompatibility (help).
      - Kanamori 2009, p. 57. Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.
      - Kanamori 2009, p. 65: "Forcing itself went a considerable distance in downgrading any formal theory of classes because of the added encumbrance of having to specify the classes of generic extensions." Kanamori, Akihiro (2009), "Bernays and Set Theory" (PDF), Bulletin of Symbolic Logic, 15 (1): 43–69, doi:10.2178/bsl/1231081769, JSTOR 25470304, S2CID 15567244.

uni-goettingen.de

gdz.sub.uni-goettingen.de

von Neumann 1925, pp. 221–224, 226, 229; English translation: van Heijenoort 2002b, pp. 396–398, 400, 403. von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
von Neumann 1925, von Neumann 1928. von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240. von Neumann, John (1928), "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift, 27: 669–752, doi:10.1007/bf01171122, S2CID 123492324.
von Neumann 1925, p. 223 (footnote); English translation: van Heijenoort 2002b, p. 398 (footnote). von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
von Neumann 1925, p. 225; English translation: van Heijenoort 2002b, p. 400. von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
von Neumann 1925, pp. 224–226; English translation: van Heijenoort 2002b, pp. 399–401. von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
von Neumann 1925, pp. 230–232; English translation: van Heijenoort 2002b, pp. 404–405. von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
von Neumann 1929, p. 229; Ferreirós 2007, pp. 379–380. von Neumann, John (1929), "Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre", Journal für die Reine und Angewandte Mathematik, 160: 227–241. Ferreirós, José (2007), Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought (2nd revised ed.), Basel, Switzerland: Birkhäuser, ISBN 978-3-7643-8349-7.

wikipedia.org

fr.wikipedia.org

Bernays' class existence axioms specify unique classes. Gödel weakened all but three of Bernays' axioms (intersection, complement, domain) by replacing biconditionals with implications, which means they specify only the ordered pairs or the 3-tuples of the class. The axioms in this section are Gödel's except for Bernays' stronger product by V axiom, which specifies a unique class of ordered pairs. Bernays' axiom simplifies the proof of the class existence theorem. Gödel's axiom B6 appears as the fourth statement of the tuple lemma. Bernays later realized that one of his axioms is redundant, which implies that one of Gödel's axioms is redundant. Using the other axioms, axiom B6 can be proved from axiom B8, and B8 can be proved from B6, so either axiom can be considered the redundant axiom.^[17] The names for the tuple-handling axioms are from the French Wikipédia article: Théorie des ensembles de von Neumann.