Set theory
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Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory — as a branch of mathematics — is mostly concerned with those that are relevant to mathematics as a whole.
The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The nonformalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the BuraliForti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the bestknown and most studied.
Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of
History
Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by
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 Bernard Bolzano in the first half of the 19th century.^{[3]} Modern understanding of infinity began in 1870–1874, and was motivated by Cantor's work in real analysis.^{[4]}
Basic concepts and notation
Set theory begins with a fundamental
A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a
Just as arithmetic features binary operations on numbers, set theory features binary operations on sets.^{[6]} The following is a partial list of them:
 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.^{[7]} For example, 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. For example, the intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3}.
 Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A. The set difference {1, 2, 3} \ {2, 3, 4} is {1}, while conversely, the set difference {2, 3, 4} \ {1, 2, 3} is {4}. When A is a subset of U, the set difference U \ A is also called the complement of A in U. In this case, if the choice of U is clear from the context, the notation A^{c} is sometimes used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams.
 Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1, 2, 3} and {2, 3, 4}, the symmetric difference set is {1, 4}. It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B) or (A \ B) ∪ (B \ A).
 Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a, b), where a is a member of A and b is a member of B. For example, the Cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.
 Power set of a set A, denoted , is the set whose members are all of the possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1, 2} }.
Some basic sets of central importance are the set of natural numbers, the set of real numbers and the empty set—the unique set containing no elements. The empty set is also occasionally called the null set,^{[8]} though this name is ambiguous and can lead to several interpretations.
Ontology
A set is
Formalized set theory
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 Russell's paradox and the BuraliForti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.^{[note 1]}
The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy.^{[a]} Such systems come in two flavors, those whose ontology consists of:
 Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Fragments of ZFC include:
 separation;
 General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
 separation and replacement.
 Sets and , both of which are stronger than ZFC.
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 New Foundations systems of NFU (allowing urelements) and NF (lacking them), associate with Willard Van Orman Quine, are not based on a cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating wellfoundedness, Thomas Forster has argued that it does reflect an iterative conception of set.^{[9]}
Systems of
An enrichment of
Applications
Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as
Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or secondorder logic. For example, properties of the natural and real numbers can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms.^{[13]}
Set theory as a foundation for
Areas of study
Set theory is a major area of research in mathematics with many interrelated subfields:
Combinatorial set theory
Combinatorial set theory concerns extensions of finite
Descriptive set theory
Descriptive set theory is the study of subsets of the
The field of
A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics.
Fuzzy set theory
In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In
Inner model theory
An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive
The study of inner models is common in the study of determinacy and large cardinals, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).^{[16]}
Large cardinals
A large cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo–Fraenkel set theory.
Determinacy
Determinacy refers to the fact that, under appropriate assumptions, certain twoplayer games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The
Forcing
Cardinal invariants
A cardinal invariant is a property of the real line measured by a cardinal number. For example, a wellstudied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.
Settheoretic topology
Settheoretic topology studies questions of general topology that are settheoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.
Controversy
From set theory's inception, some mathematicians have objected to it as a
A different objection put forth by Henri Poincaré is that defining sets using the axiom schemas of specification and replacement, as well as the axiom of power set, introduces impredicativity, a type of circularity, into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]".^{[18]}
An active area of research is the
Mathematical education
As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing the basics of naive set theory early in mathematics education.
In the US in the 1960s, the New Math experiment aimed to teach basic set theory, among other abstract concepts, to primary school students, but was met with much criticism. The math syllabus in European schools followed this trend, and currently includes the subject at different levels in all grades. Venn diagrams are widely employed to explain basic settheoretic relationships to primary school students (even though John Venn originally devised them as part of a procedure to assess the validity of inferences in term logic).
Set theory is used to introduce students to
In addition to that, sets are commonly referred to in mathematical teaching when talking about different types of numbers (the sets of
See also
 Glossary of set theory
 Class (set theory)
 List of set theory topics
 Relational model – borrows from set theory
 Venn diagram
Notes
 ^ This is the converse for ZFC; V is a model of ZFC.
References
 S2CID 199545885
 .
 ^ "Introduction to Sets", www.mathsisfun.com, retrieved 20200820
 OCLC 1527264
 ^ "set theory  Basics, Examples, & Formulas", Encyclopedia Britannica, retrieved 20200820
 ^ Bagaria, Joan (2020), "Set Theory", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Spring 2020 ed.), Metaphysics Research Lab, Stanford University, retrieved 20200820
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 ^ "6.3: Equivalence Relations and Partitions", Mathematics LibreTexts, 20191125, retrieved 20220727
 ^ "Order Relations and Functions" (PDF), Web.stanford.edu, retrieved 20220729
 Zbl 0268.26001
 ^ "A PARTITION CALCULUS IN SET THEORY" (PDF), Ams.org, retrieved 20220729
 doi:10.1101/2023.09.27.559729, retrieved 20231207)
{{cite journal}}
: Cite journal requiresjournal=
(help  ^ Rodych, Victor (Jan 31, 2018), "Wittgenstein's Philosophy of Mathematics", in Zalta, Edward N. (ed.), Stanford Encyclopedia of Philosophy (Spring 2018 ed.)
 ISBN 0631191305
 ^ Rodych 2018, §2.1: "When we prove a theorem or decide a proposition, we operate in a purely formal, syntactical manner. In doing mathematics, we do not discover preexisting truths that were 'already there without one knowing' (PG 481)—we invent mathematics, bitbylittlebit." Note, however, that Wittgenstein does not identify such deduction with philosophical logic; c.f. Rodych §1, paras. 712.
 ^ Rodych 2018, §3.4: "Given that mathematics is a 'motley of techniques of proof' (RFM III, §46), it does not require a foundation (RFM VII, §16) and it cannot be given a selfevident foundation (PR §160; WVC 34 & 62; RFM IV, §3). Since set theory was invented to provide mathematics with a foundation, it is, minimally, unnecessary."
 ^ Rodych 2018, §2.2: "An expression quantifying over an infinite domain is never a meaningful proposition, not even when we have proved, for instance, that a particular number n has a particular property."
 ^ Rodych 2018, §3.6.
 ^ homotopy type theory at the nLab
 ^ Homotopy Type Theory: Univalent Foundations of Mathematics. The Univalent Foundations Program. Institute for Advanced Study.
 ISBN 9781441174130
Further reading
 Ferreirós, Jose (2001), Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Berlin: Springer, ISBN 9783764357498
 Monk, J. Donald (1969), Introduction to Set Theory, McGrawHill Book Company, ISBN 9780898740066
 Potter, Michael (2004), Set Theory and Its Philosophy: A Critical Introduction, ISBN 9780191556432
External links
 Daniel Cunningham, Set Theory article in the Internet Encyclopedia of Philosophy.
 Jose Ferreiros, "The Early Development of Set Theory" article in the [Stanford Encyclopedia of Philosophy].
 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).
 "Axiomatic set theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 "Set theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Klein's encyclopedia.
 Online books, and library resources in your library and in other libraries about set theory