Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-22T08:45:27.547Z Has data issue: false hasContentIssue false

Set Theory

Published online by Cambridge University Press:  21 January 2022

John P. Burgess
Affiliation:
Princeton University, New Jersey

Summary

Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, controversial axioms and undecided questions, and philosophical issues raised by technical developments.
Get access
Type
Element
Information
Online ISBN: 9781108981828
Publisher: Cambridge University Press
Print publication: 10 March 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asperó, D. & Schindler, R. (2021). Martin’s Maximum++ Implies Woodin’s Axiom (*), Annals of Mathematic 193, 793835.Google Scholar
Barwise, J. (ed.) (1977). Handbook of Mathematical Logic, Amsterdam: North Holland.Google Scholar
Benacerraf, P. (1965). What Numbers Could Not Be, Philosophical Review 74, 4773. Reprinted in Benacerraf, P. & Putnam, H. (eds.) (1983). Philosophy of Mathematics: Selected Readings, 2nd ed., Englewood Cliffs: Prentice Hall (pp. 272–94).CrossRefGoogle Scholar
Benacerraf, P. & Putnam, H. (eds.) (1983). Philosophy of Mathematics: Selected Readings, 2nd ed., Englewood Cliffs: Prentice Hall.Google Scholar
Blackwell, D. (1967). Infinite Games and Analytic Sets, Proceedings of the National Academy of Sciences 58, 1836–7.CrossRefGoogle ScholarPubMed
Blumenthal, L. M. (1940). A Paradox, a Paradox, a Most Ingenious Paradox, American Mathematical Monthly 47, 346–53.Google Scholar
Boole, G. (1854). An Investigation of the Laws of Thought, London: Macmillan.Google Scholar
Boolos, G. S. (1971). The Iterative Conception of Set, Journal of Philosophy 68, 215–31. Reprinted in Benacerraf, P. & Putnam, H. (eds.) (1983). Philosophy of Mathematics: Selected Readings, 2nd ed., Englewood Cliffs: Prentice Hall (pp. 486–502).Google Scholar
Boolos, G. S., Burgess, J. P., & Jeffrey, R. C. (2002). Computability and Logic, 5th ed., Cambridge: Cambridge University Press.Google Scholar
Bourbaki, N. [collective pseud.] (1939). Théorie d’Ensembles: Fascicule de Résultats [Set Theory: Booklet of Results], Paris: Hermann.Google Scholar
Burgess, J. P. (1977). Forcing. In Barwise, J. (ed.) (1977). Handbook of Mathematical Logic, Amsterdam: North Holland (pp. 403–52).Google Scholar
Cantor, G. (1915). Contributions to the Founding of the Theory of Transfinite Numbers, tr. Jourdain, P. E. B., Chicago: Open Court.Google Scholar
Cohen, P. J. (1966). Set Theory and the Continuum Hypothesis, New York: W. A. Benjamin.Google Scholar
Davis, M. (1964). Infinite Games of Perfect Information, Annals of Mathematical Studies 52, Princeton: Princeton University Press.Google Scholar
Dedekind, R. (1901). Essays on the Theory of Numbers, tr. Beman, W. W., Chicago: Open Court.Google Scholar
Devlin, K. (1977). Constructibility. In Barwise, J. (ed.) (1977). Handbook of Mathematical Logic, Amsterdam: North Holland (pp. 453–90).Google Scholar
Erdös, P. & Tarski, A. (1961). On Some Problems Involving Inaccessible Cardinals,Google Scholar
Feferman, S., Dawson, J., & Kleene, S. (eds.) (1990). Kurt Gödel: Collected Works II, Oxford: Oxford University Press.Google Scholar
Fraenkel, A. (1922/1967). The Notion of “Definite” and the Independence of the Axiom of Choice, tr. Woodward, B. In van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press. (pp. 284–9).Google Scholar
Frege, G. (1879/1967). Begriffsschrift: A Formula Language Modeled on That of Arithmetic, for Pure Thought, tr. Bauer-Mengelberg, S. In van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press (pp. 1–82).Google Scholar
Frege, G. (1893). Grundgesetze der Arithmetik I [Basic Laws of Arithmetic I], Jena: Hermann Pohle.Google Scholar
Friedman, H. (1971). Higher Set Theory and Mathematical Practice, Annals of Mathematical Logic 2, 325–57.CrossRefGoogle Scholar
Gillman, L. (2002). Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis, American Mathematical Monthly 109, 544–53.Google Scholar
Gödel, K. (1940). The Consistency of the Continuum Hypothesis, Annals of Mathematical Studies 3, Princeton: Princeton University Press.Google Scholar
Gödel, K. (1946/1965). Remarks before the Princeton Bicentennial Conference on Problems in Mathematics. In Davis, M. (ed.) (1965). The Undecidable; Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Hewlett: Raven Press (pp. 84–8). Reprinted in Feferman, S., Dawson, J., & Kleene, S. (eds.) (1990). Kurt Gödel: Collected Works II, Oxford: Oxford University Press (pp. 150–3).Google Scholar
Gödel, K. (1947). What Is Cantor’s Continuum Problem? American Mathematical Monthly 9, 515–25. Reprinted with modifications in Benacerraf, P. & Putnam, H. (eds.) (1983). Philosophy of Mathematics: Selected Readings, 2nd ed., Englewood Cliffs: Prentice Hall (pp. 470–85), and in both original and modified versions in Feferman, S., Dawson, J., & Kleene, S. (eds.) (1990). Kurt Gödel: Collected Works II, Oxford: Oxford University Press (pp. 154–84).Google Scholar
Halmos, P. (1960). Naive Set Theory, Princeton: Van Nostrand.Google Scholar
Hamilton, W. R. (1853). Lectures on Quaternions, Dublin: Hodges & Smith.Google Scholar
Hardy, G. H. (1914). A Course of Pure Mathematics, 2nd ed., Cambridge: Cambridge University Press.Google Scholar
Hartogs, F. (1915). Über das Problem der Wohlordnung [On the Problem of Wellordering], Mathematische Annalen 36, 438–43.Google Scholar
Holmes, R. (2014). Alternative Axiomatic Set Theories, Stanford Encyclopedia of Philosophy, plato.stanford.edu/archives/fall2014/entries/settheory-alternative/.Google Scholar
Hrbacek, K. & Jech, T. (1999). Introduction to Set Theory: Revised and Expanded, 3rd ed., New York: Marcel Dekker.Google Scholar
Incurvati, L. (2020). Conceptions of Sets and Foundations of Mathematics, Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Kanamori, A. (2003). The Higher Infinite, Large Cardinals in Set Theory from Their Beginnings, Berlin, Springer.Google Scholar
Kanamori, A. (2010) (ed.). Introduction. In Foreman, M. & Kanamori, A. (eds.). Handbook of Set Theory I, Berlin: Springer (pp. 192).Google Scholar
Koellner, P. (2009). On Reflection Principles, Annals of Pure and Applied Logic 157, 206–19.Google Scholar
Kunen, K. (1970). Some Applications of Iterated Ultrapowers in Set Theory, Annals of Mathematical Logic 1, 179227.Google Scholar
Kunen, K. (1977). Combinatorics. In Barwise, J. (ed.) (1977). Handbook of Mathematical Logic, Amsterdam: North Holland (pp. 371402).Google Scholar
Kuratowski, K. (1966). Topologys, Warsaw: Polish Scientific Publishers.Google Scholar
Landau, E. (1930). Foundations of Analysis: The Arithmetic of Whole, Rational, Irrational, and Complex Numbers, tr. Steinhardt, F., Providence: Chelsea.Google Scholar
Lebesgue, H. (1902). Intégrale, Longueur, Aire [Integral, Length, Area], Milan: Bernardoni & Rebeschini.Google Scholar
Levy, A. (1960). Axiom Schemata of Strong Infinity in Axiomatic Set Theory, Pacific Journal of Mathematics 10, 223–38.Google Scholar
Maddy, P. (2011). Defending the Axioms, Oxford: Oxford University Press.Google Scholar
Maddy, P. (2017). Set-Theoretic Foundations. In Caicedo, A. E. (ed.). Foundations of Mathematics: Essays in Honor of W. Hugh Woodin’s 60th Birthday, Providence: American Mathematical Society (pp. 289322).Google Scholar
Martin, D. A. (2020) Determinacy of Infinitely Long Games (preprint). www.math.ucla.edu/~dam/booketc/thebook.pdf.Google Scholar
Martin, D. A. & Solovay, R. (1970). Internal Cohen Extensions, Annals of Mathematical Logic 2, 143–78.CrossRefGoogle Scholar
Martin, D. A. & Steel, J. R. (1989). A Proof of Projective Determinacy, Journal of the American Mathematical Society 2, 71125.CrossRefGoogle Scholar
Mathias, A. R. D. (1992). What Is Mac Lane Missing? In Judah, H., Just, W., & Woodin, H. Set Theory and the Continuum, Mathematical Sciences Research Institute Publications 26, Berlin: Springer.Google Scholar
Moschovakis, Y. (2009) Descriptive Set Theory, 2nd ed., Providence: American Mathematical Society.Google Scholar
Poincaré, H. (1905/1983). On the Nature of Mathematical Reasoning, tr. W. J. G. [initials only indicated], in Benacerraf, P. & Putnam, H. (eds.) (1983). Philosophy of Mathematics: Selected Readings, 2nd ed., Englewood Cliffs: Prentice Hall (pp. 377–93).Google Scholar
Putnam, H. (1980). Models and Reality. Journal of Symbolic Logic 45, 464–82. Reprinted in Benacerraf, P. & Putnam. H. (eds.) Philosophy of Mathematics: Selected Readings, 2nd ed., Englewood Cliffs: Prentice Hall (1983) (pp. 421–46).CrossRefGoogle Scholar
Ramsey, F. P. (1925) The Foundations of Mathematics, Proceedings of the London Mathematical Society 25, 338–84.Google Scholar
Ramsey, F. P. (1930) On a Problem of Formal Logic, Proceedings of the London Mathematical Society 30, 264–86.Google Scholar
Rubin, H. & Rubin, J. E. (1970). Equivalents of the Axiom of Choice II, Amsterdam: North Holland.Google Scholar
Rudin, M. E. (1977). Martin’s Axiom. In Barwise, J. (ed.) (1977). Handbook of Mathematical Logic, Amsterdam: North Holland (pp. 491502).Google Scholar
Russell, B. (1902). Letter to Frege. In van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press (pp. 124–5).Google Scholar
Russell, B. (1908). Mathematical Logic as Based on the Theory of Types, American Journal of Mathematics 30, 222–62. Reprinted in van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press (pp. 153–82).Google Scholar
Scott, D. (1961). Measurable Cardinals and Constructible Sets. Bulletin de l’Académie Polonaise des Sciences, Série des sciences mathématiques, astronomiques et physiques 9, 521–4.Google Scholar
Sierpinski, W. (1956). L’Hypothèse du Continu [The Continuum Hypothesis], Providence: Chelsea.Google Scholar
Sierpinski, W. (1958). Cardinal and Ordinal Numbers, Warsaw: Polish Scientific Publishers.Google Scholar
Skolem, T. (1922/1967). Some Remarks on Axiomatic Set Theory, tr. Bauer-Mengelberg, S. In van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press (pp. 290301).Google Scholar
Tarski, A. & Vaught, R. L. (1956) Arithmetical Extensions of Relational Systems, Compositio Mathematica 13, 81102.Google Scholar
van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press.Google Scholar
Vitali, G. (1905). Sulla Problema della Mesura dei Gruppi di Punti di una Retta [On the Problem of the Measure of Sets of Points on a Line], Bologna: Gamberini & Parmeggiani.Google Scholar
von Neumann, J. (1923/1967). On the Introduction of Transfinite Numbers, tr. Bauer-Mengelberg, S. In van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press (pp. 346–54).Google Scholar
Wachover, N. (2021). How Many Numbers Exist? Infinite Proof Moves Math Closer to an Answer, Quanta Magazine. www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/.Google Scholar
Whitehead, A. N. & Russell, B. (1910). Principia Mathematical I, Cambridge: Cambridge University Press.Google Scholar
Zermelo, E. (1908/1967). Investigations in the Foundations of Set Theory I, tr. Bauer-Mengelberg, S. In van Heijenoort, J. (1967). From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Cambridge, MA: Harvard University Press (pp. 199–215).Google Scholar
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre [On Boundary-Numbers and Set-Domains: New Investigations in the Foundations of Set Theory], Fundamenta Mathematicae 16, 2947.Google Scholar

Save element to Kindle

To save this element to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Set Theory
  • John P. Burgess, Princeton University, New Jersey
  • Online ISBN: 9781108981828
Available formats
×

Save element to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Set Theory
  • John P. Burgess, Princeton University, New Jersey
  • Online ISBN: 9781108981828
Available formats
×

Save element to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Set Theory
  • John P. Burgess, Princeton University, New Jersey
  • Online ISBN: 9781108981828
Available formats
×