English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2.
English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2. Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, North-Holland, ISBN0-444-86839-9.
Zermelo used this sequence of cardinals to obtain a sequence of models that explains the paradoxes of set theory — such as, the Burali-Forti paradox and Russell's paradox. He stated that the paradoxes "depend solely on confusing set theory itself ... with individual models representing it. What appears as an 'ultrafinite non- or super-set' in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain [model]." (Zermelo 1930, pp. 46–47; English translation: Ewald 1996, p. 1223.) 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.
English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2.
"... we must, on the one hand, restrict these principles [axioms] sufficiently to exclude all contradictions and, on the other hand, take them sufficiently wide to retain all that is valuable in this theory." (Zermelo 1908, p. 261; English translation: van Heijenoort 1967a, p. 200). Gregory Moore argues that Zermelo's "axiomatization was primarily motivated by a desire to secure his demonstration of the Well-Ordering Theorem ..." (Moore 1982, pp. 158–160). Zermelo, Ernst (1908), "Untersuchungen über die Grundlagen der Mengenlehre", Mathematische Annalen, 65 (2): 261–281, doi:10.1007/bf01449999, S2CID120085563.
English translation: van Heijenoort, Jean (1967a), "Investigations in the foundations of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 199–215, ISBN978-0-674-32449-7.
English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2.
English translation: van Heijenoort, Jean (1967c), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413, ISBN978-0-674-32449-7. Levy, Azriel (1968), "On Von Neumann's Axiom System for Set Theory", American Mathematical Monthly, 75 (7): 762–763, doi:10.2307/2315201, JSTOR2315201.
English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2.
English translation: van Heijenoort, Jean (1967b), "Some remarks on axiomatized set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 290–301, ISBN978-0-674-32449-7.
English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2.
English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2. Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, North-Holland, ISBN0-444-86839-9.
Zermelo used this sequence of cardinals to obtain a sequence of models that explains the paradoxes of set theory — such as, the Burali-Forti paradox and Russell's paradox. He stated that the paradoxes "depend solely on confusing set theory itself ... with individual models representing it. What appears as an 'ultrafinite non- or super-set' in one model is, in the succeeding model, a perfectly good, valid set with both a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain [model]." (Zermelo 1930, pp. 46–47; English translation: Ewald 1996, p. 1223.) 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.
English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2.
English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2.
English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN978-0-19-853271-2.
English translation: van Heijenoort, Jean (1967c), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413, ISBN978-0-674-32449-7. Levy, Azriel (1968), "On Von Neumann's Axiom System for Set Theory", American Mathematical Monthly, 75 (7): 762–763, doi:10.2307/2315201, JSTOR2315201.
"... we must, on the one hand, restrict these principles [axioms] sufficiently to exclude all contradictions and, on the other hand, take them sufficiently wide to retain all that is valuable in this theory." (Zermelo 1908, p. 261; English translation: van Heijenoort 1967a, p. 200). Gregory Moore argues that Zermelo's "axiomatization was primarily motivated by a desire to secure his demonstration of the Well-Ordering Theorem ..." (Moore 1982, pp. 158–160). Zermelo, Ernst (1908), "Untersuchungen über die Grundlagen der Mengenlehre", Mathematische Annalen, 65 (2): 261–281, doi:10.1007/bf01449999, S2CID120085563.
English translation: van Heijenoort, Jean (1967a), "Investigations in the foundations of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 199–215, ISBN978-0-674-32449-7.
English translation: van Heijenoort, Jean (1967b), "Some remarks on axiomatized set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 290–301, ISBN978-0-674-32449-7.
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gdz.sub.uni-goettingen.de
"... we must, on the one hand, restrict these principles [axioms] sufficiently to exclude all contradictions and, on the other hand, take them sufficiently wide to retain all that is valuable in this theory." (Zermelo 1908, p. 261; English translation: van Heijenoort 1967a, p. 200). Gregory Moore argues that Zermelo's "axiomatization was primarily motivated by a desire to secure his demonstration of the Well-Ordering Theorem ..." (Moore 1982, pp. 158–160). Zermelo, Ernst (1908), "Untersuchungen über die Grundlagen der Mengenlehre", Mathematische Annalen, 65 (2): 261–281, doi:10.1007/bf01449999, S2CID120085563.
English translation: van Heijenoort, Jean (1967a), "Investigations in the foundations of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 199–215, ISBN978-0-674-32449-7.
Von Neumann investigated whether his set theory is categorical; that is, whether it uniquely determines sets in the sense that any two of its models are isomorphic. He showed that it is not categorical because of a weakness in the axiom of regularity: this axiom only excludes descending ∈-sequences from existing in the model; descending sequences may still exist outside the model. A model having "external" descending sequences is not isomorphic to a model having no such sequences since this latter model lacks isomorphic images for the sets belonging to external descending sequences. This led von Neumann to conclude "that no categorical axiomatization of set theory seems to exist at all" (von Neumann 1925, p. 239; English translation: van Heijenoort 1967c, p. 412). von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
English translation: van Heijenoort, Jean (1967c), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413, ISBN978-0-674-32449-7.
English translation: van Heijenoort, Jean (1967c), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413, ISBN978-0-674-32449-7.
English translation: van Heijenoort, Jean (1967c), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413, ISBN978-0-674-32449-7.
English translation: van Heijenoort, Jean (1967c), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413, ISBN978-0-674-32449-7. Levy, Azriel (1968), "On Von Neumann's Axiom System for Set Theory", American Mathematical Monthly, 75 (7): 762–763, doi:10.2307/2315201, JSTOR2315201.
English translation: van Heijenoort, Jean (1967c), "An axiomatization of set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 393–413, ISBN978-0-674-32449-7.
English translation: van Heijenoort, Jean (1967b), "Some remarks on axiomatized set theory", From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, pp. 290–301, ISBN978-0-674-32449-7.