Axiom of limitation of size (English Wikipedia)

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

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Bernays 1937, pp. 66–70; Bernays 1941, pp. 1–6. Gödel 1940, pp. 3–7. Kelley 1955, pp. 251–273. 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, 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 Continuum Hypothesis, Princeton University Press. Kelley, John L. (1955), General Topology, Van Nostrand, ISBN 978-0-387-90125-1 {{citation}}: ISBN / Date incompatibility (help).
Fraenkel, Bar-Hillel & Levy 1973, p. 137. Fraenkel, Abraham; Bar-Hillel, Yehoshua; Levy, Azriel (1973), Foundations of Set Theory (2nd revised ed.), Basel, Switzerland: Elsevier, ISBN 0-7204-2270-1.
Fraenkel, Bar-Hillel & Levy 1973, pp. 32, 137. Fraenkel, Abraham; Bar-Hillel, Yehoshua; Levy, Azriel (1973), Foundations of Set Theory (2nd revised ed.), Basel, Switzerland: Elsevier, ISBN 0-7204-2270-1.
Fraenkel, Bar-Hillel & Levy 1973, p. 95. Fraenkel, Abraham; Bar-Hillel, Yehoshua; Levy, Azriel (1973), Foundations of Set Theory (2nd revised ed.), Basel, Switzerland: Elsevier, ISBN 0-7204-2270-1.

bu.edu

math.bu.edu

From a Nov. 8, 1957 letter Gödel wrote to Stanislaw Ulam (Kanamori 2003, p. 295). Kanamori, Akihiro (2003), "Stanislaw Ulam" (PDF), in Solomon Feferman and John W. Dawson, Jr. (ed.), Kurt Gödel Collected Works, Volume V: Correspondence H-Z, Clarendon Press, pp. 280–300, ISBN 0-19-850075-0.

doi.org

This is the standard definition of V₀. Zermelo let V₀ be a set of urelements and proved that if this set contains a single element, the resulting model satisfies the axiom of limitation of size (his proof also works for V₀ = ∅). Zermelo stated that the axiom is not true for all models built from a set of urelements. (Zermelo 1930, p. 38; English translation: Ewald 1996, p. 1227.) 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.
- This is Zermelo's definition (Zermelo 1930, p. 36; English translation: Ewald 1996, p. 1225.). If V₀ = ∅, this definition is equivalent to the standard definition V_α+1 = P(V_α) since V_α ⊆ P(V_α) (Kunen 1980, p. 95; Kunen uses the notation R(α) instead of V_α). If V₀ is a set of urelements, the standard definition eliminates the urelements at V₁. 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.
  - Zermelo introduced strongly inaccessible cardinals κ so that V_κ would satisfy ZFC. The axioms of power set and replacement led him to the properties of strongly inaccessible cardinals. (Zermelo 1930, pp. 31–35; English translation: Ewald 1996, pp. 1221–1224.) Independently, Wacław Sierpiński and Alfred Tarski introduced these cardinals in 1930. (Sierpiński & Tarski 1930.) 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.
    - 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.
      - "... 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, S2CID 120085563.
        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, ISBN 978-0-674-32449-7.
        Von Neumann published an introductory article on his axiom system in 1925 (von Neumann 1925; English translation: van Heijenoort 1967c). In 1928, he provided a detailed treatment of his system (von Neumann 1928). 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, ISBN 978-0-674-32449-7. von Neumann, John (1928), "Die Axiomatisierung der Mengenlehre", Mathematische Zeitschrift, 27: 669–752, doi:10.1007/bf01171122, S2CID 123492324.
        Bernays 1937, pp. 66–70; Bernays 1941, pp. 1–6. Gödel 1940, pp. 3–7. Kelley 1955, pp. 251–273. 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, 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 Continuum Hypothesis, Princeton University Press. Kelley, John L. (1955), General Topology, Van Nostrand, ISBN 978-0-387-90125-1 {{citation}}: ISBN / Date incompatibility (help).
        Zermelo 1930; English translation: Ewald 1996. 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, ISBN 978-0-19-853271-2.
        Levy 1968. Levy, Azriel (1968), "On Von Neumann's Axiom System for Set Theory", American Mathematical Monthly, 75 (7): 762–763, doi:10.2307/2315201, JSTOR 2315201.
        It came 43 years later: von Neumann stated his axioms in 1925 and Lévy's proof appeared in 1968. (von Neumann 1925, Levy 1968.) 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, ISBN 978-0-674-32449-7. Levy, Azriel (1968), "On Von Neumann's Axiom System for Set Theory", American Mathematical Monthly, 75 (7): 762–763, doi:10.2307/2315201, JSTOR 2315201.
        Gödel 1939, p. 223. Gödel, Kurt (1939), "Consistency Proof for the Generalized Continuum Hypothesis" (PDF), Proceedings of the National Academy of Sciences of the United States of America, 25 (4): 220–224, Bibcode:1939PNAS...25..220G, doi:10.1073/pnas.25.4.220, PMC 1077751, PMID 16588293.
        These theorems are part of Zermelo's Second Development Theorem. (Zermelo 1930, p. 37; English translation: Ewald 1996, p. 1226.) 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, ISBN 978-0-19-853271-2.
        Fraenkel 1922, pp. 230–231. Skolem 1922; English translation: van Heijenoort 1967b, pp. 296–297). Fraenkel, Abraham (1922), "Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre", Mathematische Annalen, 86 (3–4): 230–237, doi:10.1007/bf01457986, S2CID 122212740. Skolem, Thoralf (1922), "Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre", Matematikerkongressen i Helsingfors den 4-7 Juli, 1922, pp. 217–232.
        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, ISBN 978-0-674-32449-7.

harvard.edu

ui.adsabs.harvard.edu

Gödel 1939, p. 223. Gödel, Kurt (1939), "Consistency Proof for the Generalized Continuum Hypothesis" (PDF), Proceedings of the National Academy of Sciences of the United States of America, 25 (4): 220–224, Bibcode:1939PNAS...25..220G, doi:10.1073/pnas.25.4.220, PMC 1077751, PMID 16588293.

icm.edu.pl

matwbn.icm.edu.pl

This is the standard definition of V₀. Zermelo let V₀ be a set of urelements and proved that if this set contains a single element, the resulting model satisfies the axiom of limitation of size (his proof also works for V₀ = ∅). Zermelo stated that the axiom is not true for all models built from a set of urelements. (Zermelo 1930, p. 38; English translation: Ewald 1996, p. 1227.) 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.
- This is Zermelo's definition (Zermelo 1930, p. 36; English translation: Ewald 1996, p. 1225.). If V₀ = ∅, this definition is equivalent to the standard definition V_α+1 = P(V_α) since V_α ⊆ P(V_α) (Kunen 1980, p. 95; Kunen uses the notation R(α) instead of V_α). If V₀ is a set of urelements, the standard definition eliminates the urelements at V₁. 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.
  - Zermelo introduced strongly inaccessible cardinals κ so that V_κ would satisfy ZFC. The axioms of power set and replacement led him to the properties of strongly inaccessible cardinals. (Zermelo 1930, pp. 31–35; English translation: Ewald 1996, pp. 1221–1224.) Independently, Wacław Sierpiński and Alfred Tarski introduced these cardinals in 1930. (Sierpiński & Tarski 1930.) 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.
    - 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.
      - Zermelo 1930; English translation: Ewald 1996. 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, ISBN 978-0-19-853271-2.
        These theorems are part of Zermelo's Second Development Theorem. (Zermelo 1930, p. 37; English translation: Ewald 1996, p. 1226.) 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, ISBN 978-0-19-853271-2.

jstor.org

Bernays 1937, pp. 66–70; Bernays 1941, pp. 1–6. Gödel 1940, pp. 3–7. Kelley 1955, pp. 251–273. 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, 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 Continuum Hypothesis, Princeton University Press. Kelley, John L. (1955), General Topology, Van Nostrand, ISBN 978-0-387-90125-1 {{citation}}: ISBN / Date incompatibility (help).
Levy 1968. Levy, Azriel (1968), "On Von Neumann's Axiom System for Set Theory", American Mathematical Monthly, 75 (7): 762–763, doi:10.2307/2315201, JSTOR 2315201.
It came 43 years later: von Neumann stated his axioms in 1925 and Lévy's proof appeared in 1968. (von Neumann 1925, Levy 1968.) von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.

nih.gov

ncbi.nlm.nih.gov

Gödel 1939, p. 223. Gödel, Kurt (1939), "Consistency Proof for the Generalized Continuum Hypothesis" (PDF), Proceedings of the National Academy of Sciences of the United States of America, 25 (4): 220–224, Bibcode:1939PNAS...25..220G, doi:10.1073/pnas.25.4.220, PMC 1077751, PMID 16588293.

pubmed.ncbi.nlm.nih.gov

Gödel 1939, p. 223. Gödel, Kurt (1939), "Consistency Proof for the Generalized Continuum Hypothesis" (PDF), Proceedings of the National Academy of Sciences of the United States of America, 25 (4): 220–224, Bibcode:1939PNAS...25..220G, doi:10.1073/pnas.25.4.220, PMC 1077751, PMID 16588293.

pnas.org

Gödel 1939, p. 223. Gödel, Kurt (1939), "Consistency Proof for the Generalized Continuum Hypothesis" (PDF), Proceedings of the National Academy of Sciences of the United States of America, 25 (4): 220–224, Bibcode:1939PNAS...25..220G, doi:10.1073/pnas.25.4.220, PMC 1077751, PMID 16588293.

semanticscholar.org

api.semanticscholar.org

"... 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, S2CID 120085563.
- Von Neumann published an introductory article on his axiom system in 1925 (von Neumann 1925; English translation: van Heijenoort 1967c). In 1928, he provided a detailed treatment of his system (von Neumann 1928). von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
  - Bernays 1937, pp. 66–70; Bernays 1941, pp. 1–6. Gödel 1940, pp. 3–7. Kelley 1955, pp. 251–273. 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, 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 Continuum Hypothesis, Princeton University Press. Kelley, John L. (1955), General Topology, Van Nostrand, ISBN 978-0-387-90125-1 {{citation}}: ISBN / Date incompatibility (help).
  - Fraenkel 1922, pp. 230–231. Skolem 1922; English translation: van Heijenoort 1967b, pp. 296–297). Fraenkel, Abraham (1922), "Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre", Mathematische Annalen, 86 (3–4): 230–237, doi:10.1007/bf01457986, S2CID 122212740. Skolem, Thoralf (1922), "Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre", Matematikerkongressen i Helsingfors den 4-7 Juli, 1922, pp. 217–232.

uni-goettingen.de

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, S2CID 120085563.
- Von Neumann published an introductory article on his axiom system in 1925 (von Neumann 1925; English translation: van Heijenoort 1967c). In 1928, he provided a detailed treatment of his system (von Neumann 1928). von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
  - 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.
    - For example, von Neumann's proof that his axiom implies the well-ordering theorem uses the Burali-Forte paradox (von Neumann 1925, p. 223; English translation: van Heijenoort 1967c, p. 398). von Neumann, John (1925), "Eine Axiomatisierung der Mengenlehre", Journal für die Reine und Angewandte Mathematik, 154: 219–240.
      - von Neumann 1925, p. 223; English translation: van Heijenoort 1967c, pp. 397–398. 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, ISBN 978-0-674-32449-7.
        It came 43 years later: von Neumann stated his axioms in 1925 and Lévy's proof appeared in 1968. (von Neumann 1925, Levy 1968.) 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, ISBN 978-0-674-32449-7. Levy, Azriel (1968), "On Von Neumann's Axiom System for Set Theory", American Mathematical Monthly, 75 (7): 762–763, doi:10.2307/2315201, JSTOR 2315201.
        von Neumann 1925, p. 223; English translation: van Heijenoort 1967c, p. 398. Von Neumann's proof, which only uses axioms, has the advantage of applying to all models rather than just to V_κ. 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, ISBN 978-0-674-32449-7.
        Fraenkel 1922, pp. 230–231. Skolem 1922; English translation: van Heijenoort 1967b, pp. 296–297). Fraenkel, Abraham (1922), "Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre", Mathematische Annalen, 86 (3–4): 230–237, doi:10.1007/bf01457986, S2CID 122212740. Skolem, Thoralf (1922), "Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre", Matematikerkongressen i Helsingfors den 4-7 Juli, 1922, pp. 217–232.
        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, ISBN 978-0-674-32449-7.

worldcat.org

search.worldcat.org

Zermelo introduced strongly inaccessible cardinals κ so that V_κ would satisfy ZFC. The axioms of power set and replacement led him to the properties of strongly inaccessible cardinals. (Zermelo 1930, pp. 31–35; English translation: Ewald 1996, pp. 1221–1224.) Independently, Wacław Sierpiński and Alfred Tarski introduced these cardinals in 1930. (Sierpiński & Tarski 1930.) 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.