Axiom of choice (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Axiom of choice" in English language version.

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Fraenkel, Bar-Hillel & Lévy 1973, pp. 69–70. Fraenkel, Abraham A.; Bar-Hillel, Yehoshua; Lévy, Azriel (1973). Foundations of set theory (2nd ed.). Amsterdam-London: North-Holland Publishing Co. pp. 69–70. ISBN 9780080887050. MR 0345816.
Blass, Andreas (1984). "Existence of bases implies the axiom of choice". Axiomatic set theory (Boulder, Colo., 1983). Contemporary Mathematics. Vol. 31. Providence, RI: American Mathematical Society. pp. 31–33. doi:10.1090/conm/031/763890. ISBN 978-0-8218-5026-8. MR 0763890.
Serre, Jean-Pierre (2003), Trees, Springer Monographs in Mathematics, Springer, p. 23; Soukup, Lajos (2008), "Infinite combinatorics: from finite to infinite", Horizons of combinatorics, Bolyai Society Mathematical Studies, vol. 17, Berlin: Springer, pp. 189–213, CiteSeerX 10.1.1.222.5699, doi:10.1007/978-3-540-77200-2_10, ISBN 978-3-540-77199-9, MR 2432534. See in particular Theorem 2.1, pp. 192–193.

archive.org

Herrlich 2006, p. 9. According to Suppes 1972, p. 243, this was the formulation of the axiom of choice which was originally given by Zermelo 1904. See also Halmos 1960, p. 60 for this formulation. Herrlich, Horst (2006). Axiom of Choice. Lecture Notes in Math. 1876. Berlin: Springer-Verlag. ISBN 978-3-540-30989-5. Suppes, Patrick (1972) [1960]. Axiomatic set theory. Mineola, New York: Dover. ISBN 978-0-486-61630-8. Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann" (reprint). Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300. S2CID 124189935. Halmos, Paul R. (1960). Naive Set Theory. The University Series in Undergraduate Mathematics. Princeton, NJ: van Nostrand Company. Zbl 0087.04403.
Suppes 1972, p. 240. Suppes, Patrick (1972) [1960]. Axiomatic set theory. Mineola, New York: Dover. ISBN 978-0-486-61630-8.
See Moore 2013, pp. 330–334, for a structured list of 74 equivalents. See Howard & Rubin 1998, pp. 11–16, for 86 equivalents with source references. Moore, Gregory H (2013) [1982]. Zermelo's axiom of choice: Its origins, development & influence. Mineola, New York: Dover Publications. ISBN 978-0-486-48841-7. Howard, Paul; Rubin, Jean E. (1998), Consequences of the axiom of choice, Mathematical Surveys and Monographs, vol. 59, Providence, Rhode Island: American Mathematical Society, ISBN 9780821809778
Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. pp. 20–24. ISBN 978-0199237180. OCLC 740446073.

books.google.com

Rosenberg 2021. Rosenberg, Steven (21 December 2021). An Invitation to Abstract Algebra. CRC Press. ISBN 9781000516333.
Fraenkel, Bar-Hillel & Lévy 1973, pp. 69–70. Fraenkel, Abraham A.; Bar-Hillel, Yehoshua; Lévy, Azriel (1973). Foundations of set theory (2nd ed.). Amsterdam-London: North-Holland Publishing Co. pp. 69–70. ISBN 9780080887050. MR 0345816.
Rosenbloom 2005, p. 147. Rosenbloom, Paul C. (2005). The Elements of Mathematical Logic. Courier Dover Publications. ISBN 9780486446172.
Serre, Jean-Pierre (2003), Trees, Springer Monographs in Mathematics, Springer, p. 23; Soukup, Lajos (2008), "Infinite combinatorics: from finite to infinite", Horizons of combinatorics, Bolyai Society Mathematical Studies, vol. 17, Berlin: Springer, pp. 189–213, CiteSeerX 10.1.1.222.5699, doi:10.1007/978-3-540-77200-2_10, ISBN 978-3-540-77199-9, MR 2432534. See in particular Theorem 2.1, pp. 192–193.

cmu.edu

cs.cmu.edu

Per Martin-Löf, Intuitionistic type theory, 1980. Anne Sjerp Troelstra, Metamathematical investigation of intuitionistic arithmetic and analysis, Springer, 1973.

doi.org

Zermelo 1904. Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann" (reprint). Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300. S2CID 124189935.
Herrlich 2006, p. 9. According to Suppes 1972, p. 243, this was the formulation of the axiom of choice which was originally given by Zermelo 1904. See also Halmos 1960, p. 60 for this formulation. Herrlich, Horst (2006). Axiom of Choice. Lecture Notes in Math. 1876. Berlin: Springer-Verlag. ISBN 978-3-540-30989-5. Suppes, Patrick (1972) [1960]. Axiomatic set theory. Mineola, New York: Dover. ISBN 978-0-486-61630-8. Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann" (reprint). Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300. S2CID 124189935. Halmos, Paul R. (1960). Naive Set Theory. The University Series in Undergraduate Mathematics. Princeton, NJ: van Nostrand Company. Zbl 0087.04403.
Dawson 2006: "The axiom of choice, though it had been employed unconsciously in many arguments in analysis, became controversial once made explicit, not only because of its non-constructive character, but because it implied such extremely unintuitive consequences as the Banach–Tarski paradox." Dawson, J. W. (August 2006). "Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödel's Impact on Logic, Mathematics, and Computer Science". Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science (LICS 2006). pp. 339–341. doi:10.1109/LICS.2006.47. ISBN 978-0-7695-2631-7. S2CID 15526447.
Martin-Löf, Per (2006). "100 Years of Zermelo's Axiom of Choice: What was the Problem with It?". The Computer Journal. 49 (3): 345–350. Bibcode:1980CompJ..23..262L. doi:10.1093/comjnl/bxh162.
Gödel, Kurt (9 November 1938). "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 24 (12): 556–557. Bibcode:1938PNAS...24..556G. doi:10.1073/pnas.24.12.556. PMC 1077160. PMID 16577857.
Blass, Andreas (1984). "Existence of bases implies the axiom of choice". Axiomatic set theory (Boulder, Colo., 1983). Contemporary Mathematics. Vol. 31. Providence, RI: American Mathematical Society. pp. 31–33. doi:10.1090/conm/031/763890. ISBN 978-0-8218-5026-8. MR 0763890.
Blass 1979. Blass, Andreas (1979). "Injectivity, Projectivity and the Axiom of Choice". Transactions of the American Mathematical Society. 255: 31–59. doi:10.2307/1998165. JSTOR 1998165.
Serre, Jean-Pierre (2003), Trees, Springer Monographs in Mathematics, Springer, p. 23; Soukup, Lajos (2008), "Infinite combinatorics: from finite to infinite", Horizons of combinatorics, Bolyai Society Mathematical Studies, vol. 17, Berlin: Springer, pp. 189–213, CiteSeerX 10.1.1.222.5699, doi:10.1007/978-3-540-77200-2_10, ISBN 978-3-540-77199-9, MR 2432534. See in particular Theorem 2.1, pp. 192–193.
It was shown by Lévy 1958 and others using Mostowski models that eight definitions of a finite set are independent in ZF without AC, although they are equivalent when AC is assumed. The definitions are I-finite, Ia-finite, II-finite, III-finite, IV-finite, V-finite, VI-finite and VII-finite. I-finiteness is the same as normal finiteness. IV-finiteness is the same as Dedekind-finiteness. Lévy, Azriel (1958). "The independence of various definitions of finiteness" (PDF). Fundamenta Mathematicae. 46: 1–13. doi:10.4064/fm-46-1-1-13. Archived (PDF) from the original on 9 October 2022.
Sageev, Gershon (March 1975). "An independence result concerning the axiom of choice". Annals of Mathematical Logic. 8 (1–2): 1–184. doi:10.1016/0003-4843(75)90002-9.
Ash, C. J. (1975). "A consequence of the axiom of choice". Journal of the Australian Mathematical Society. 19 (3): 306–308. doi:10.1017/S1446788700031505. S2CID 122334025.
Stavi, Jonathan (1974). "A model of ZF with an infinite free complete Boolean algebra". Israel Journal of Mathematics. 20 (2): 149–163. doi:10.1007/BF02757883. S2CID 119543439.

ghostarchive.org

Cohen, Paul (2019). "The Independence of the Axiom of Choice" (PDF). Stanford University Libraries. Archived (PDF) from the original on 9 October 2022. Retrieved 22 March 2019.
It was shown by Lévy 1958 and others using Mostowski models that eight definitions of a finite set are independent in ZF without AC, although they are equivalent when AC is assumed. The definitions are I-finite, Ia-finite, II-finite, III-finite, IV-finite, V-finite, VI-finite and VII-finite. I-finiteness is the same as normal finiteness. IV-finiteness is the same as Dedekind-finiteness. Lévy, Azriel (1958). "The independence of various definitions of finiteness" (PDF). Fundamenta Mathematicae. 46: 1–13. doi:10.4064/fm-46-1-1-13. Archived (PDF) from the original on 9 October 2022.

harvard.edu

ui.adsabs.harvard.edu

Martin-Löf, Per (2006). "100 Years of Zermelo's Axiom of Choice: What was the Problem with It?". The Computer Journal. 49 (3): 345–350. Bibcode:1980CompJ..23..262L. doi:10.1093/comjnl/bxh162.
Gödel, Kurt (9 November 1938). "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 24 (12): 556–557. Bibcode:1938PNAS...24..556G. doi:10.1073/pnas.24.12.556. PMC 1077160. PMID 16577857.

icm.edu.pl

matwbn.icm.edu.pl

It was shown by Lévy 1958 and others using Mostowski models that eight definitions of a finite set are independent in ZF without AC, although they are equivalent when AC is assumed. The definitions are I-finite, Ia-finite, II-finite, III-finite, IV-finite, V-finite, VI-finite and VII-finite. I-finiteness is the same as normal finiteness. IV-finiteness is the same as Dedekind-finiteness. Lévy, Azriel (1958). "The independence of various definitions of finiteness" (PDF). Fundamenta Mathematicae. 46: 1–13. doi:10.4064/fm-46-1-1-13. Archived (PDF) from the original on 9 October 2022.

jstor.org

Blass 1979. Blass, Andreas (1979). "Injectivity, Projectivity and the Axiom of Choice". Transactions of the American Mathematical Society. 255: 31–59. doi:10.2307/1998165. JSTOR 1998165.

karagila.org

"On the Partition Principle".

ncatlab.org

projective object at the nLab

nih.gov

ncbi.nlm.nih.gov

Gödel, Kurt (9 November 1938). "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 24 (12): 556–557. Bibcode:1938PNAS...24..556G. doi:10.1073/pnas.24.12.556. PMC 1077160. PMID 16577857.

pubmed.ncbi.nlm.nih.gov

Gödel, Kurt (9 November 1938). "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. 24 (12): 556–557. Bibcode:1938PNAS...24..556G. doi:10.1073/pnas.24.12.556. PMC 1077160. PMID 16577857.

nyu.edu

cs.nyu.edu

"[FOM] Are (C,+) and (R,+) isomorphic". 21 February 2006.

psu.edu

citeseerx.ist.psu.edu

Serre, Jean-Pierre (2003), Trees, Springer Monographs in Mathematics, Springer, p. 23; Soukup, Lajos (2008), "Infinite combinatorics: from finite to infinite", Horizons of combinatorics, Bolyai Society Mathematical Studies, vol. 17, Berlin: Springer, pp. 189–213, CiteSeerX 10.1.1.222.5699, doi:10.1007/978-3-540-77200-2_10, ISBN 978-3-540-77199-9, MR 2432534. See in particular Theorem 2.1, pp. 192–193.

semanticscholar.org

api.semanticscholar.org

Zermelo 1904. Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann" (reprint). Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300. S2CID 124189935.
Herrlich 2006, p. 9. According to Suppes 1972, p. 243, this was the formulation of the axiom of choice which was originally given by Zermelo 1904. See also Halmos 1960, p. 60 for this formulation. Herrlich, Horst (2006). Axiom of Choice. Lecture Notes in Math. 1876. Berlin: Springer-Verlag. ISBN 978-3-540-30989-5. Suppes, Patrick (1972) [1960]. Axiomatic set theory. Mineola, New York: Dover. ISBN 978-0-486-61630-8. Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann" (reprint). Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300. S2CID 124189935. Halmos, Paul R. (1960). Naive Set Theory. The University Series in Undergraduate Mathematics. Princeton, NJ: van Nostrand Company. Zbl 0087.04403.
Dawson 2006: "The axiom of choice, though it had been employed unconsciously in many arguments in analysis, became controversial once made explicit, not only because of its non-constructive character, but because it implied such extremely unintuitive consequences as the Banach–Tarski paradox." Dawson, J. W. (August 2006). "Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödel's Impact on Logic, Mathematics, and Computer Science". Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science (LICS 2006). pp. 339–341. doi:10.1109/LICS.2006.47. ISBN 978-0-7695-2631-7. S2CID 15526447.
Ash, C. J. (1975). "A consequence of the axiom of choice". Journal of the Australian Mathematical Society. 19 (3): 306–308. doi:10.1017/S1446788700031505. S2CID 122334025.
Stavi, Jonathan (1974). "A model of ZF with an infinite free complete Boolean algebra". Israel Journal of Mathematics. 20 (2): 149–163. doi:10.1007/BF02757883. S2CID 119543439.

stanford.edu

plato.stanford.edu

John, Bell (10 December 2021). "The Axiom of Choice". Stanford Encyclopedia of Philosophy. Retrieved 2 December 2024. Let us call Zermelo's 1908 formulation the combinatorial axiom of choice: CAC: Any collection of mutually disjoint nonempty sets has a transversal.
"Quine's New Foundations". Stanford Encyclopedia of Philosophy. Retrieved 10 November 2017.

stacks.stanford.edu

Cohen, Paul (2019). "The Independence of the Axiom of Choice" (PDF). Stanford University Libraries. Archived (PDF) from the original on 9 October 2022. Retrieved 22 March 2019.

uni-goettingen.de

gdz.sub.uni-goettingen.de

Zermelo 1904. Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann" (reprint). Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300. S2CID 124189935.
Herrlich 2006, p. 9. According to Suppes 1972, p. 243, this was the formulation of the axiom of choice which was originally given by Zermelo 1904. See also Halmos 1960, p. 60 for this formulation. Herrlich, Horst (2006). Axiom of Choice. Lecture Notes in Math. 1876. Berlin: Springer-Verlag. ISBN 978-3-540-30989-5. Suppes, Patrick (1972) [1960]. Axiomatic set theory. Mineola, New York: Dover. ISBN 978-0-486-61630-8. Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann" (reprint). Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300. S2CID 124189935. Halmos, Paul R. (1960). Naive Set Theory. The University Series in Undergraduate Mathematics. Princeton, NJ: van Nostrand Company. Zbl 0087.04403.

worldcat.org

search.worldcat.org

Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. pp. 20–24. ISBN 978-0199237180. OCLC 740446073.
Schechter 1996, pp. 391–392. Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.

zbmath.org

Herrlich 2006, p. 9. According to Suppes 1972, p. 243, this was the formulation of the axiom of choice which was originally given by Zermelo 1904. See also Halmos 1960, p. 60 for this formulation. Herrlich, Horst (2006). Axiom of Choice. Lecture Notes in Math. 1876. Berlin: Springer-Verlag. ISBN 978-3-540-30989-5. Suppes, Patrick (1972) [1960]. Axiomatic set theory. Mineola, New York: Dover. ISBN 978-0-486-61630-8. Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann" (reprint). Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300. S2CID 124189935. Halmos, Paul R. (1960). Naive Set Theory. The University Series in Undergraduate Mathematics. Princeton, NJ: van Nostrand Company. Zbl 0087.04403.
Fraenkel 1922. Fraenkel, Abraham (1922), "Der Begriff "definit" und die Unabhängigkeit des Auswahlaxioms", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften: 253–257, JFM 48.0199.02