Contents 1 History 2 Basic concepts and notation 3 Some ontology 4 Axiomatic set theory 5 Applications 6 Areas of study 6.1 Combinatorial set theory
6.2 Descriptive set theory
6.3
7 Objections to set theory as a foundation for mathematics 8 See also 9 Notes 10 Further reading 11 External links History[edit] Georg Cantor. Mathematical topics typically emerge and evolve through interactions
among many researchers. Set theory, however, was founded by a single
paper in 1874 by Georg Cantor: "On a Property of the Collection of All
Real Algebraic Numbers".[1][2]
Since the 5th century BC, beginning with Greek mathematician Zeno of
Elea in the West and early Indian mathematicians in the East,
mathematicians had struggled with the concept of infinity. Especially
notable is the work of
Union of the sets A and B, denoted A ∪ B, is the set of all objects
that are a member of A, or B, or both. The union of 1, 2, 3 and 2,
3, 4 is the set 1, 2, 3, 4 .
Intersection of the sets A and B, denoted A ∩ B, is the set of all
objects that are members of both A and B. The intersection of 1, 2, 3
and 2, 3, 4 is the set 2, 3 .
Some basic sets of central importance are the empty set (the unique set containing no elements; occasionally called the null set though this name is ambiguous), the set of natural numbers, and the set of real numbers. Some ontology[edit] Main article: von Neumann universe An initial segment of the von Neumann hierarchy. A set is pure if all of its members are sets, all members of its
members are sets, and so on. For example, the set containing only
the empty set is a nonempty pure set. In modern set theory, it is
common to restrict attention to the von Neumann universe of pure sets,
and many systems of axiomatic set theory are designed to axiomatize
the pure sets only. There are many technical advantages to this
restriction, and little generality is lost, because essentially all
mathematical concepts can be modeled by pure sets. Sets in the von
Neumann universe are organized into a cumulative hierarchy, based on
how deeply their members, members of members, etc. are nested. Each
set in this hierarchy is assigned (by transfinite recursion) an
ordinal number α, known as its rank. The rank of a pure set X is
defined to be the least upper bound of all successors of ranks of
members of X. For example, the empty set is assigned rank 0, while the
set containing only the empty set is assigned rank 1. For each
ordinal α, the set Vα is defined to consist of all pure sets with
rank less than α. The entire von Neumann universe is denoted V.
Axiomatic set theory[edit]
Elementary set theory can be studied informally and intuitively, and
so can be taught in primary schools using Venn diagrams. The intuitive
approach tacitly assumes that a set may be formed from the class of
all objects satisfying any particular defining condition. This
assumption gives rise to paradoxes, the simplest and best known of
which are
Sets alone. This includes the most common axiomatic set theory,
Sets and proper classes. These include Von Neumann–Bernays–Gödel
set theory, which has the same strength as
The above systems can be modified to allow urelements, objects that
can be members of sets but that are not themselves sets and do not
have any members.
The systems of
Glossary of set theory
Notes[edit] ^ 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
^ Johnson, Philip (1972), A History of Set Theory, Prindle, Weber
& Schmidt, ISBN 0-87150-154-6
^ 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
^
Further reading[edit] Devlin, Keith, 1993. The Joy of Sets (2nd ed.). Springer Verlag,
ISBN 0-387-94094-4
Ferreirós, Jose, 2007 (1999). Labyrinth of Thought: A history of set
theory and its role in modern mathematics. Basel, Birkhäuser.
ISBN 978-3-7643-8349-7
Johnson, Philip, 1972. A History of Set Theory. Prindle, Weber &
Schmidt ISBN 0-87150-154-6
Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence
Proofs. North-Holland, ISBN 0-444-85401-0.
Potter, Michael, 2004.
External links[edit] Wikibooks has a book on the topic of: Set Theory Wikibooks has a book on the topic of: Discrete mathematics/Set theory Foreman, Matthew, Akihiro Kanamori, eds. Handbook of Set Theory. 3 vols., 2010. Each chapter surveys some aspect of contemporary research in set theory. Does not cover established elementary set theory, on which see Devlin (1993). Hazewinkel, Michiel, ed. (2001) [1994], "Axiomatic set theory", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Hazewinkel, Michiel, ed. (2001) [1994], "Set theory", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 Jech, Thomas (2002). "Set Theory", Stanford Encyclopedia of Philosophy. Schoenflies, Arthur (1898). Mengenlehre in Klein's encyclopedia. Online books, and library resources in your library and in other libraries about set theory v t e Set theory Axioms Choice countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom
replacement specification Operations Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union Concepts Methods Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram Set types Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset · Superset Transitive Uncountable Universal Theories Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine v t e Areas of mathematics outline topic lists Branches Arithmetic History of mathematics
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