Set Theory in Mathematics
Keywords:
community, mathematics, Contemporary researchAbstract
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as the Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
References
Jump up^ Cantor, Georg (1874), "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", J. Reine Angew. Math., 77: 258–262, doi:10.1515/crll.1874.77.258 2. Jump up^ Johnson, Philip (1972), A History of Set Theory, Prindle, Weber & Schmidt, ISBN 0-87150-154-6 3. Jump up^ Bolzano, Bernard (1975), Berg, Jan, ed., Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre, Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., Vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich Frommann Verlag, p. 152, ISBN 3-7728-0466-7 4. Jump up^ William Henry Young & Grace Chisholm Young (1906) Theory of Sets of Points, link from Internet Archive 5. Jump up^ In his 1925, John von Neumann observed that "set theory in its first, "naive" version, due to Cantor, led to contradictions. These are the well-known antinomies of the set of all sets that do not contain themselves (Russell), of the set of all transfinte ordinal numbers (Burali-Forti), and the set of all finitely definable real numbers (Richard)." He goes on to observe that two "tendencies" were attempting to "rehabilitate" set theory. Of the first effort, exemplified by Bertrand Russell, Julius König, Hermann Weyl and L. E. J. Brouwer, von Neumann called the "overall effect of their activity . . . devastating". With regards to the axiomatic method employed by second group composed of Zermelo, Abraham Fraenkel and Arthur Moritz Schoenflies, von Neumann worried that "We see only that the known modes of inference leading to the antinomies fail, but who knows where there are not others?" and he set to the task, "in the spirit of the second group", to "produce, by means of a finite number of purely formal operations . . . all the sets that we want to see formed" but not allow for the antinomies. (All quotes from von Neumann 1925 reprinted in van Heijenoort, Jean (1967, third printing 1976), "From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931", Harvard University Press, Cambridge MA, ISBN 0-674-32449-8 (pbk). A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann's 1925. 6. Jump up^ Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (Third Millennium ed.), Berlin, New York: Springer-Verlag, p. 642, ISBN 978-3-540-44085-7, Zbl 1007.03002 7. Jump up^ Bishop, Errett 1967. Foundations of Constructive Analysis, New York: Academic Press. ISBN 4-87187-714-0 8. Jump up^ Solomon Feferman, 1998, In the Light of Logic, Oxford Univ. Press (New York), p.280-283 and 293-294 9. Jump up^ Wittgenstein, Ludwig (1975). Philosophical Remarks, §129, §174. Oxford: Basil Blackwell. ISBN 0631191305. 10. Jump up^ Ferro, A.; Omodeo, E. G.; Schwartz, J. T. (1980), "Decision procedures for elementary sublanguages of set theory. I. Multi-level syllogistic and some extensions", Comm. Pure Appl. Math., 33 (5): 599–608, doi:10.1002/cpa.3160330503 11. Jump up^ Saunders Mac Lane and Ieke Moerdijk (1992) Sheaves in Geometry and Logic: a First Introduction to Topos Theory. Springer Verlag. 12. Jump up^ homotopy type theory in nLab 13. Jump up^ Homotopy Type Theory: Univalent Foundations of Mathematics. The Univalent Foundations Program. Institute for Advanced Study.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2017 International Journal for Research Publication and Seminar
This work is licensed under a Creative Commons Attribution 4.0 International License.
Re-users must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. This license allows for redistribution, commercial and non-commercial, as long as the original work is properly credited.
