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54-600: Richard Joseph Laver (October 20, 1942 – September 19, 2012) was an American mathematician, working in set theory . Laver received his PhD at the University of California, Berkeley in 1969, under the supervision of Ralph McKenzie , with a thesis on Order Types and Well-Quasi-Orderings . The largest part of his career he spent as Professor and later Emeritus Professor at the University of Colorado at Boulder . Richard Laver died in Boulder, CO , on September 19, 2012 after

108-550: A mathematical function as a relation from one set (the domain ) to another set (the range ). Subset When quantified, A ⊆ B {\displaystyle A\subseteq B} is represented as ∀ x ( x ∈ A ⇒ x ∈ B ) . {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).} One can prove

162-616: 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 well-foundedness, Thomas Forster has argued that it does reflect an iterative conception of set . Systems of constructive set theory , such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic . Yet other systems accept classical logic but feature

216-574: 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

270-422: A larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models . A cardinal invariant

324-401: A long illness. Among Laver's notable achievements some are the following. Set theory 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

378-485: A model V of ZF satisfies the continuum hypothesis or the axiom of choice , the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent. The study of inner models is common in the study of determinacy and large cardinals , especially when considering axioms such as

432-476: A natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone spaces . An active area of research is the univalent foundations and related to it homotopy type theory . Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types . Principles such as

486-712: A nonstandard membership relation. These include rough set theory and fuzzy set theory , in which the value of an atomic formula embodying the membership relation is not simply True or False . The Boolean-valued models of ZFC are a related subject. An enrichment of ZFC called internal set theory was proposed by Edward Nelson in 1977. Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs , manifolds , rings , vector spaces , and relational algebras can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and

540-455: A set A . If o is a member (or element ) of A , the notation o ∈ A is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets is the subset relation, also called set inclusion . If all

594-849: A spectacular blunder in Remarks on the Foundations of Mathematics : Wittgenstein attempted to refute Gödel's incompleteness theorems after having only read the abstract. As reviewers Kreisel , Bernays , Dummett , and Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as Crispin Wright begun to rehabilitate Wittgenstein's arguments. Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism , finite set theory, and computable set theory. Topoi also give

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648-514: A vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals . 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 ". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in

702-486: 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 non-formalized 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

756-514: Is less than y (an irreflexive relation ). Similarly, using the convention that ⊂ {\displaystyle \subset } is proper subset, if A ⊆ B , {\displaystyle A\subseteq B,} then A may or may not equal B , but if A ⊂ B , {\displaystyle A\subset B,} then A definitely does not equal B . Another example in an Euler diagram : The set of all subsets of S {\displaystyle S}

810-486: Is vacuously a subset of any set X . Some authors use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols ⊆ {\displaystyle \subseteq } and ⊇ . {\displaystyle \supseteq .} For example, for these authors, it

864-530: 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 refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from

918-443: Is a partial list of them: 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 , though this name is ambiguous and can lead to several interpretations. The Power set of a set A , denoted P ( A ) {\displaystyle {\mathcal {P}}(A)} ,

972-741: Is a property of the real line measured by a cardinal number. For example, a well-studied 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. Set-theoretic topology studies questions of general topology that are set-theoretic 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

1026-480: Is a subset of B , but A is not equal to B . Also, 1, 2, and 3 are members (elements) of the set {1, 2, 3} , but are not subsets of it; and in turn, the subsets, such as {1} , are not members of the set {1, 2, 3} . More complicated relations can exist; for example, the set {1} is both a member and a proper subset of the set {1, {1}} . Just as arithmetic features binary operations on numbers , set theory features binary operations on sets. The following

1080-546: Is between set theory and recursion theory . It includes the study of lightface pointclasses , and is closely related to hyperarithmetical theory . In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable. A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations . This has important applications to

1134-803: Is called its power set , and is denoted by P ( S ) {\displaystyle {\mathcal {P}}(S)} . The inclusion relation ⊆ {\displaystyle \subseteq } is a partial order on the set P ( S ) {\displaystyle {\mathcal {P}}(S)} defined by A ≤ B ⟺ A ⊆ B {\displaystyle A\leq B\iff A\subseteq B} . We may also partially order P ( S ) {\displaystyle {\mathcal {P}}(S)} by reverse set inclusion by defining A ≤ B  if and only if  B ⊆ A . {\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.} For

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1188-439: Is denoted  V {\displaystyle V} . 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

1242-533: Is equivalent to A ⊆ B , {\displaystyle A\subseteq B,} as stated above. If A and B are sets and every element of A is also an element of B , then: If A is a subset of B , but A is not equal to B (i.e. there exists at least one element of B which is not an element of A ), then: The empty set , written { } {\displaystyle \{\}} or ∅ , {\displaystyle \varnothing ,} has no elements, and therefore

1296-476: Is more flexible than a simple yes or no answer and can be a real number such as 0.75. An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether

1350-491: Is strictly greater than the rank of any of its elements. 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 {\displaystyle \alpha } , the set V α {\displaystyle V_{\alpha }} is defined to consist of all pure sets with rank less than α {\displaystyle \alpha } . The entire von Neumann universe

1404-676: 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]". Ludwig Wittgenstein condemned set theory philosophically for its connotations of mathematical platonism . He wrote that "set theory

1458-418: 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. From set theory's inception, some mathematicians have objected to it as a foundation for mathematics . The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from

1512-400: 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} } . Notably, P ( A ) {\displaystyle {\mathcal {P}}(A)} contains both A and the empty set. 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

1566-405: Is true of every set A that A ⊂ A . {\displaystyle A\subset A.} (a reflexive relation ). Other authors prefer to use the symbols ⊂ {\displaystyle \subset } and ⊃ {\displaystyle \supset } to indicate proper (also called strict) subset and proper superset respectively; that is, with

1620-702: Is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein identified mathematics with algorithmic human deduction; the need for a secure foundation for mathematics seemed, to him, nonsensical. Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical constructivism and finitism . Meta-mathematical statements — which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory — are not mathematics. Few modern philosophers have adopted Wittgenstein's views after

1674-454: The Axiom of Choice have recently seen applications in evolutionary dynamics , enhancing the understanding of well-established models of evolution and interaction. Set theory is a major area of research in mathematics with many interrelated subfields: Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and

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1728-510: The Burali-Forti 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 best-known 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

1782-601: The Burali-Forti paradox . Axiomatic set theory was originally devised to rid set theory of such paradoxes. The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy . Such systems come in two flavors, those whose ontology consists of: 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

1836-496: The axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results. As set theory gained popularity as

1890-506: The constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop 's influential book Foundations of Constructive Analysis . A different objection put forth by Henri Poincaré

1944-500: The k -tuple from { 0 , 1 } k , {\displaystyle \{0,1\}^{k},} of which the i th coordinate is 1 if and only if s i {\displaystyle s_{i}} is a member of T . The set of all k {\displaystyle k} -subsets of A {\displaystyle A} is denoted by ( A k ) {\displaystyle {\tbinom {A}{k}}} , in analogue with

1998-404: 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. Set theory as a foundation for mathematical analysis , topology , abstract algebra , and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from

2052-504: 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. Modern understanding of infinity began in 1870–1874, and was motivated by Cantor's work in real analysis . Set theory begins with a fundamental binary relation between an object o and

2106-402: 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). A large cardinal

2160-399: 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

2214-502: The framework to develop a mathematical theory of infinity , and has various applications in computer science (such as in the theory of relational algebra ), philosophy , formal semantics , and evolutionary dynamics . Its foundational appeal, together with its paradoxes , and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for logicians and philosophers of mathematics . Contemporary research into set theory covers

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2268-690: The letter A "), which may be useful when learning computer programming , since Boolean logic is used in various programming languages . Likewise, sets and other collection-like objects, such as multisets and lists , are common datatypes in computer science and programming . In addition to that, sets are commonly referred to in mathematical teaching when talking about different types of numbers (the sets N {\displaystyle \mathbb {N} } of natural numbers , Z {\displaystyle \mathbb {Z} } of integers , R {\displaystyle \mathbb {R} } of real numbers , etc.), and when defining

2322-444: The members of set A are also members of set B , then A is a subset of B , denoted A ⊆ B . For example, {1, 2} is a subset of {1, 2, 3} , and so is {2} but {1, 4} is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A

2376-1000: The power set P ⁡ ( S ) {\displaystyle \operatorname {\mathcal {P}} (S)} of a set S , the inclusion partial order is—up to an order isomorphism —the Cartesian product of k = | S | {\displaystyle k=|S|} (the cardinality of S ) copies of the partial order on { 0 , 1 } {\displaystyle \{0,1\}} for which 0 < 1. {\displaystyle 0<1.} This can be illustrated by enumerating S = { s 1 , s 2 , … , s k } , {\displaystyle S=\left\{s_{1},s_{2},\ldots ,s_{k}\right\},} , and associating with each subset T ⊆ S {\displaystyle T\subseteq S} (i.e., each element of 2 S {\displaystyle 2^{S}} )

2430-492: The real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure. Paul Cohen invented the method of forcing while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additional sets in order to create

2484-507: The relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath , includes human-written, computer-verified derivations of more than 12,000 theorems starting from ZFC set theory, first-order logic and propositional logic . ZFC and

2538-736: The same meaning as and instead of the symbols ⊊ {\displaystyle \subsetneq } and ⊋ . {\displaystyle \supsetneq .} This usage makes ⊆ {\displaystyle \subseteq } and ⊂ {\displaystyle \subset } analogous to the inequality symbols ≤ {\displaystyle \leq } and < . {\displaystyle <.} For example, if x ≤ y , {\displaystyle x\leq y,} then x may or may not equal y , but if x < y , {\displaystyle x<y,} then x definitely does not equal y , and

2592-439: 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 axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of

2646-917: The statement A ⊆ B {\displaystyle A\subseteq B} by applying a proof technique known as the element argument : Let sets A and B be given. To prove that A ⊆ B , {\displaystyle A\subseteq B,} The validity of this technique can be seen as a consequence of universal generalization : the technique shows ( c ∈ A ) ⇒ ( c ∈ B ) {\displaystyle (c\in A)\Rightarrow (c\in B)} for an arbitrarily chosen element c . Universal generalisation then implies ∀ x ( x ∈ A ⇒ x ∈ B ) , {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),} which

2700-401: The study of invariants in many fields of mathematics. In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people"

2754-772: The study of extensions of Ramsey's theorem such as the Erdős–Rado theorem . Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces . It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy . Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals. The field of effective descriptive set theory

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2808-478: The subject at different levels in all grades. Venn diagrams are widely employed to explain basic set-theoretic 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 logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition ) of sets (e.g. "months starting with

2862-456: The theory of mathematical relations can be described in set theory. 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 second-order logic . For example, properties of

2916-420: 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 {\displaystyle \alpha } , known as its rank. The rank of a pure set X {\displaystyle X} is defined to be the least ordinal that

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