66-518: The MICRO Relational Database Management System was the first large-scale set-theoretic database management system to be used in production. Though MICRO was initially considered to be an "Information Management System", it was eventually recognized to provide all the capabilities of an RDBMS. MICRO's major underpinnings and algorithms were based on the Set-Theoretic Data Structure (STDS) model developed by D. L. Childs of
132-443: A mathematical function as a relation from one set (the domain ) to another set (the range ). Richard Dedekind Julius Wilhelm Richard Dedekind ( German: [ˈdeːdəˌkɪnt] ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory , abstract algebra (particularly ring theory ), and the axiomatic foundations of arithmetic . His best known contribution
198-497: A one-to-one correspondence between them. He invoked similarity to give the first precise definition of an infinite set : a set is infinite when it is "similar to a proper part of itself," in modern terminology, is equinumerous to one of its proper subsets . Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N , ( N → N ): Dedekind's work in this area anticipated that of Georg Cantor , who
264-616: 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
330-574: 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
396-422: 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
462-537: A matrix or table in which the rows represent different records or "cases", and the columns contain individual data items for each record; however, the actual data representation is in set-theoretic form. In labor market applications the rows typically represent job applicants or employees and columns represent fields such as age, sex, and income or type of industry, number of employees, and payroll. MICRO permits users with little programming experience to define, enter, interrogate, manipulate, and update collections of data in
528-485: 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
594-476: 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
660-712: 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
726-780: A professor at the Collegium. Richard Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. His body rests at Braunschweig Main Cemetery . He first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind
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#1732771816147792-405: A relatively unstructured and unconstrained environment. An interactive system, MICRO is powerful in terms of the complexity of requests which can be made by users without prior programming language experience. MICRO includes basic statistical computations such as mean, variance, frequency, median, etc. If more rigorous statistical analysis are desired, the data from a MICRO database can be exported to
858-455: 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
924-849: 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
990-612: A subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of Emmy Noether . Ideals generalize Ernst Eduard Kummer 's ideal numbers , devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem . (Thus Dedekind can be said to have been Kummer's most important disciple.) In an 1882 article, Dedekind and Heinrich Martin Weber applied ideals to Riemann surfaces , giving an algebraic proof of
1056-514: 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
1122-486: 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
1188-530: 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
1254-443: 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)} ,
1320-741: 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
1386-480: 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
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#17327718161471452-546: 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
1518-610: Is commonly considered the founder of set theory . Likewise, his contributions to the foundations of mathematics anticipated later works by major proponents of logicism , such as Gottlob Frege and Bertrand Russell . Dedekind edited the collected works of Lejeune Dirichlet , Gauss , and Riemann . Dedekind's study of Lejeune Dirichlet's work led him to his later study of algebraic number fields and ideals . In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie ("Lectures on Number Theory") about which it has been written that: Although
1584-439: 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
1650-476: 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
1716-491: 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
1782-676: 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
1848-418: 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
1914-524: Is the definition of real numbers through the notion of Dedekind cut . He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as logicism . Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig . His mother was Caroline Henriette Dedekind (née Emperius), the daughter of
1980-400: 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
2046-702: 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
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2112-454: 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
2178-510: 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
2244-601: 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
2310-548: The Michigan Terminal System (MTS), the interactive time-sharing system developed at the University of Michigan that runs on IBM System/360 Model 67 , System/370 , and compatible mainframe computers. MICRO provides a query language, a database directory, and a data dictionary to create an interface between the user and the very efficient proprietary Set-Theoretic Data Structure (STDS) software developed by
2376-539: The Riemann–Roch theorem . In 1888, he published a short monograph titled Was sind und was sollen die Zahlen? ("What are numbers and what are they good for?" Ewald 1996: 790), which included his definition of an infinite set . He also proposed an axiomatic foundation for the natural numbers, whose primitive notions were the number one and the successor function . The next year, Giuseppe Peano , citing Dedekind, formulated an equivalent but simpler set of axioms , now
2442-636: The University of Berlin , not Göttingen , was the main facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries; they were both awarded the habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent , giving courses on probability and geometry . He studied for a while with Peter Gustav Lejeune Dirichlet , and they became good friends. Because of lingering weaknesses in his mathematical knowledge, he studied elliptic and abelian functions . Yet he
2508-496: 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
2574-506: 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
2640-404: 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
2706-661: The square root of 2 defines all the nonnegative numbers whose squares are less than 2 and the negative numbers into the lesser class, and the positive numbers whose squares are greater than 2 into the greater class. Every location on the number line continuum contains either a rational or an irrational number. Thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers"); in modern terminology, Vollständigkeit , completeness . Dedekind defined two sets to be "similar" when there exists
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2772-636: The Michigan Interactive Data Analysis System (MIDAS), a statistical analysis package available under the Michigan Terminal System. Set-theoretic 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
2838-599: The Set-Theoretic Information Systems Corporation (STIS) of Ann Arbor, Michigan. The lower level routines from STIS treat the data bases as sets and perform set operations on them, e.g., union, intersection, restrictions, etc. Although the underlying STDS model is based on set theory, the MICRO user interface is similar to those subsequently used in relational database management systems . MICRO's data representation can be thought of as
2904-638: The U.S. Department of Labor. MICRO was first used for the study of large social science data bases referred to as micro data; hence the name. Organizations such as the US Department of Labor , the US Environmental Protection Agency , and researchers from the University of Alberta , the University of Michigan , Wayne State University , the University of Newcastle upon Tyne , and Durham University used MICRO to manage very large scale databases until 1998. MICRO runs under
2970-564: The University of Michigan's CONCOMP (Conversational Use of Computers) Project. MICRO featured a natural language interface which allowed non-programmers to use the system. Implementation of MICRO began in 1970 as part of the Labor Market Information System (LMIS) project at the University of Michigan's Institute of Labor and Industrial Relations (ILIR). Dr. Malcolm S. Cohen was Director of the LMIS Project and
3036-504: 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
3102-402: 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
3168-600: The book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death. The 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory . (The word "Ring", introduced later by Hilbert , does not appear in Dedekind's work.) Dedekind defined an ideal as
3234-399: 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
3300-467: The first time at the Polytechnic school, Dedekind developed the notion now known as a Dedekind cut (German: Schnitt ), now a standard definition of the real numbers. The idea of a cut is that an irrational number divides the rational numbers into two classes ( sets ), with all the numbers of one class (greater) being strictly greater than all the numbers of the other (lesser) class. For example,
3366-502: 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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#17327718161473432-690: 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
3498-444: 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
3564-492: 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
3630-507: 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
3696-595: The rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish. He never married, instead living with his sister Julia. Dedekind was elected to the Academies of Berlin (1880) and Rome, and to the French Academy of Sciences (1900). He received honorary doctorates from the universities of Oslo , Zurich , and Braunschweig . While teaching calculus for
3762-557: The standard ones. Dedekind made other contributions to algebra . For instance, around 1900, he wrote the first papers on modular lattices . In 1872, while on holiday in Interlaken , Dedekind met Georg Cantor . Thus began an enduring relationship of mutual respect, and Dedekind became one of the first mathematicians to admire Cantor's work concerning infinite sets, proving a valued ally in Cantor's disputes with Leopold Kronecker , who
3828-439: 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
3894-401: 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"
3960-772: 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
4026-478: 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
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#17327718161474092-456: 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
4158-420: 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
4224-516: Was also the first at Göttingen to lecture concerning Galois theory . About this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic . In 1858, he began teaching at the Polytechnic school in Zürich (now ETH Zürich). When the Collegium Carolinum was upgraded to a Technische Hochschule (Institute of Technology) in 1862, Dedekind returned to his native Braunschweig, where he spent
4290-464: Was taught number theory by professor Moritz Stern . Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale ("On the Theory of Eulerian integrals "). This thesis did not display the talent evident in Dedekind's subsequent publications. At that time,
4356-455: Was the principal innovator and designer of the original MICRO Retrieval System. Carol Easthope and Jack Guskin were the principal programmers. D.L. Childs, Vice President of Set Theoretic Information Systems (STIS) Corporation, provided continuing guidance in the use of Set-Theoretic Data Structure (STDS) data access software for MICRO. Funding came from the Office of Manpower Administration within
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