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In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy classifies …

An illustration of how the levels of the hierarchy interracts and where some basic sets category lies within it.

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1. Cartesian product – In Set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B, products can be specified using set-builder notation, e. g. A table can be created by taking the Cartesian product of a set of rows, If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More generally, a Cartesian product of n sets, also known as an n-fold Cartesian product, can be represented by an array of n dimensions, an ordered pair is a 2-tuple or couple. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, an illustrative example is the standard 52-card deck. The standard playing card ranks form a 13-element set, the card suits form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form, both sets are distinct, even disjoint. The main historical example is the Cartesian plane in analytic geometry, usually, such a pairs first and second components are called its x and y coordinates, respectively, cf. picture. The set of all such pairs is thus assigned to the set of all points in the plane, a formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, the Kuratowski definition, is =, note that, under this definition, X × Y ⊆ P, where P represents the power set. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, let A, B, C, and D be sets. × C ≠ A × If for example A =, then × A = ≠ = A ×, the Cartesian product behaves nicely with respect to intersections, cf. left picture. × = ∩ In most cases the above statement is not true if we replace intersection with union, cf. middle picture. Other properties related with subsets are, if A ⊆ B then A × C ⊆ B × C, the cardinality of a set is the number of elements of the set. For example, defining two sets, A = and B =, both set A and set B consist of two elements each. Their Cartesian product, written as A × B, results in a new set which has the following elements, each element of A is paired with each element of B. Each pair makes up one element of the output set, the number of values in each element of the resulting set is equal to the number of sets whose cartesian product is being taken,2 in this case

2. Stephen Cole Kleene – Stephen Cole Kleene /ˈkleɪniː/ KLAY-nee was an American mathematician. Kleenes work grounds the study of functions are computable. A number of concepts are named after him, Kleene hierarchy, Kleene algebra, the Kleene star, Kleenes recursion theorem. He also invented regular expressions, and made significant contributions to the foundations of mathematical intuitionism, although his last name is commonly pronounced /ˈkliːniː/ KLEE-nee or /ˈkliːn/ kleen, Kleene himself pronounced it /ˈkleɪniː/ KLAY-nee. His son, Ken Kleene, wrote, As far as I am aware this pronunciation is incorrect in all known languages, I believe that this novel pronunciation was invented by my father. Kleene was awarded the BA degree from Amherst College in 1930 and he was awarded the Ph. D. in mathematics from Princeton University in 1934. His thesis, entitled A Theory of Positive Integers in Formal Logic, was supervised by Alonzo Church, in the 1930s, he did important work on Churchs lambda calculus. In 1935, he joined the department at the University of Wisconsin–Madison. After two years as an instructor, he was appointed assistant professor in 1937, while a visiting scholar at the Institute for Advanced Study in Princeton, 1939–40, he laid the foundation for recursion theory, an area that would be his lifelong research interest. In 1941, he returned to Amherst College, where he spent one year as a professor of mathematics. During World War II, Kleene was a lieutenant commander in the United States Navy. He was an instructor of navigation at the U. S. Naval Reserves Midshipmens School in New York, in 1946, Kleene returned to Wisconsin, becoming a full professor in 1948 and the Cyrus C. MacDuffee professor of mathematics in 1964 and he was chair of the Department of Mathematics and Computer Science, 1962–63, and Dean of the College of Letters and Science from 1969 to 1974. The latter appointment he took on despite the considerable student unrest of the day and he retired from the University of Wisconsin in 1979. In 1999 the mathematics library at the University of Wisconsin was renamed in his honor, Kleenes teaching at Wisconsin resulted in three texts in mathematical logic, Kleene and Kleene and Vesley, often cited and still in print. Kleene wrote alternative proofs to the Gödels incompleteness theorems that enhanced their status and made them easier to teach. Kleene and Vesley is the classic American introduction to intuitionist logic, Kleene served as president of the Association for Symbolic Logic, 1956–58, and of the International Union of History and Philosophy of Science,1961. In 1990, he was awarded the National Medal of Science, the importance of Kleenes work led to the saying that Kleeneness is next to Gödelness

3. Andrzej Mostowski – Andrzej Mostowski was a Polish mathematician. He is perhaps best remembered for the Mostowski collapse lemma, born in Lemberg, Austria-Hungary, Mostowski entered University of Warsaw in 1931. He was influenced by Kuratowski, Lindenbaum and Tarski and his Ph. D. came in 1939, officially directed by Kuratowski but in practice directed by Tarski who was a young lecturer at that time. He became an accountant after the German invasion of Poland but continued working in the Underground Warsaw University, after the Warsaw uprising of 1944 the Nazis tried to put him in a concentration camp. With the help of some Polish nurses he escaped to a hospital, some of this research he reconstructed after the War, however much of it remained lost. This work was largely on recursion theory and undecidability, from 1946 until his death in Vancouver, British Columbia, Canada, he worked at the University of Warsaw. Much of work during that time was on first order logic and his son Tadeusz is also a mathematician working on differential geometry. With Krzysztof Kurdyka and Adam Parusinski, Tadeusz Mostowski solved René Thoms gradient conjecture in 2000, list of Polish People Mostowski model Kuratowski, Kazimierz, Mostowski, Andrzej, Set theory. With an introduction to set theory, Studies in Logic. Andrzej Mostowski, Über die Unabhängigkeit des Wohlordnungssatzes von Ordnungsprinzip, Fundamenta Mathematicae Vol.32, No.1, ss. Andrzej Mostowski, On definable sets of integers, Fundamenta Mathematicae Vol.34. Andrzej Mostowski, Un théorème sur les nombres cos 2πk/n, Colloquium Mathematicae Vol.1, casimir Kuratowski, Andrzej Mostowski, Sur un problème de la théorie des groupes et son rapport à la topologie, Colloquium Mathematicae Vol.2, No. Andrzej Mostowski, Groups connected with Boolean algebras, Andrzej Mostowski, On direct products of theories, Journal of Symbolic Logic, Vol.17, No. Andrzej Mostowski, Models of axiomatic systems, Fundamenta Mathematicae Vol.39, Andrzej Mostowski, On a system of axioms which has no recursively enumerable arithmetic model, Fundamenta Mathematicae Vol.40, No. Andrzej Mostowski, A formula with no recursively enumerable model, Fundamenta Mathematicae Vol.42, Andrzej Mostowski, Examples of sets definable by means of two and three quantifiers, Fundamenta Mathematicae Vol.42, No. Andrzej Mostowski, Contributions to the theory of sets and functions, Fundamenta Mathematicae Vol.42. Andrzej Ehrenfeucht, Andrzej Mostowski, Models of Axiomatic Theories Admitting Automorphisms, Fundamenta Mathematicae, Vol.43, Andrzej Mostowski, Loeuvre scientifique de Jan Łukasiewicz dans le domaine de la logique mathématique, Fundamenta Mathematicae Vol.44, No. Andrzej Mostowski, On a generalization of quantifiers, Fundamenta Mathematicae Vol.44, Andrzej Mostowski, On computable sequences, Fundamenta Mathematicae Vol.44, No