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König's Lemma König's lemma König's lemma or Kőnig's infinity lemma is a theorem in graph theory due to Dénes Kőnig (1927).[1][2] It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic, especially in computability theory. This theorem also has important roles in constructive mathematics and proof theory.Contents1 Statement of the lemma1.1 Proof2 Computability aspects 3 Relationship to constructive mathematics and compactness 4 Relationship with the axiom of choice 5 See also 6 Notes 7 References 8 External linksStatement of the lemma[edit] Let G be a connected, locally finite, infinite graph (this means, in particular, that each vertex is connected to only finitely many other vertices) [...More...]  "König's Lemma" on: Wikipedia Yahoo Parouse 

Basis Theorem (computability) In computability theory, there are a number of basis theorems. These theorems show that particular kinds of sets always must have some members that are, in terms of Turing degree, not too complicated. One family of basis theorems concern nonempty effectively closed sets (that is, nonempty Π 1 0 displaystyle Pi _ 1 ^ 0 sets in the arithmetical hierarchy); these theorems are studied as part of classical computability theory. Another family of basis theorems concern nonempty lightface analytic sets (that is, Σ 1 1 displaystyle Sigma _ 1 ^ 1 in the analytical hierarchy); these theorems are studied as part of hyperarithmetical theory.Contents1 Effectively closed sets 2 Lightface analytic sets 3 References 4 External linksEffectively closed sets[edit] Effectively closed sets are a topic of study in classical computability theory [...More...]  "Basis Theorem (computability)" on: Wikipedia Yahoo Parouse 

Luitzen Egbertus Jan Brouwer Luitzen Egbertus Jan Brouwer ForMemRS[1] (/ˈbraʊ.ər/; Dutch: [ˈlœy̯tsə(n) ɛɣˈbɛrtəs jɑn ˈbrʌu̯ər]; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and complex analysis.[2][4][5] He was the founder of the mathematical philosophy of intuitionism.Contents1 Biography 2 Bibliography2.1 In English translation3 See also 4 References 5 Further reading 6 External linksBiography[edit] Early in his career, Brouwer proved a number of theorems that were in the emerging field of topology. The main results were his fixed point theorem, the topological invariance of degree, and the topological invariance of dimension. The most popular of the three among mathematicians is the first one called the Brouwer Fixed Point Theorem [...More...]  "Luitzen Egbertus Jan Brouwer" on: Wikipedia Yahoo Parouse 

Theorem In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.[2] Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises [...More...]  "Theorem" on: Wikipedia Yahoo Parouse 

Halting Problem In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running or continue to run forever. Alan Turing Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible programinput pairs cannot exist. A key part of the proof was a mathematical definition of a computer and program, which became known as a Turing machine; the halting problem is undecidable over Turing machines. It is one of the first examples of a decision problem. Informally, for any program f that might determine if programs halt, a "pathological" program g called with an input can pass its own source and its input to f and then specifically do the opposite of what f predicts g will do [...More...]  "Halting Problem" on: Wikipedia Yahoo Parouse 

Low (computability) In computability theory, a Turing degree [X] is low if the Turing jump [X′] is 0′. A set is low if it has low degree. Since every set is computable from its jump, any low set is computable in 0′, but the jump of sets computable in 0′ can bound any degree r.e. in 0′ (Schoenfield Jump Inversion). X being low says that its jump X′ has the least possible degree in terms of Turing reducibility for the jump of a set. A degree is low n if its n'th jump is the n'th jump of 0. A set X is generalized low if it satisfies X′ ≡T X + 0′, that is: if its jump has the lowest degree possible [...More...]  "Low (computability)" on: Wikipedia Yahoo Parouse 

Low Basis Theorem The low basis theorem is one of several basis theorems in computability theory, each of which shows that, given an infinite subtree of the binary tree 2 < ω displaystyle 2^ <omega , it is possible to find an infinite path through the tree with particular computability properties. The low basis theorem, in particular, shows that there must be a path which is low, that is, the Turing jump of the path is Turing equivalent to the halting problem ∅ ′ displaystyle emptyset ' . Statement and proof[edit] The low basis theorem states that every nonempty Π 1 0 displaystyle Pi _ 1 ^ 0 class in 2 ω displaystyle 2^ omega (see arithmetical hierarchy) contains a set of low degree (Soare 1987:109) [...More...]  "Low Basis Theorem" on: Wikipedia Yahoo Parouse 

Secondorder Arithmetic In mathematical logic, secondorder arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. It was introduced by David Hilbert and Paul Bernays Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of secondorder arithmetic is denoted Z2. Secondorder arithmetic includes, but is significantly stronger than, its firstorder counterpart Peano arithmetic. Unlike Peano arithmetic, secondorder arithmetic allows quantification over sets of natural numbers as well as numbers themselves. Because real numbers can be represented as (infinite) sets of natural numbers in wellknown ways, and because second order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in secondorder arithmetic [...More...]  "Secondorder Arithmetic" on: Wikipedia Yahoo Parouse 

Robert I. Soare Robert Irving Soare is an American mathematician. He is the Paul Snowden Russell Distinguished Service Professor of Mathematics and Computer Science at the University of Chicago, where he has been on the faculty since 1967. He proved, together with Carl Jockusch, the low basis theorem, and has done other work in mathematical logic, primarily in the area of computability theory. In 2012 he became a fellow of the American Mathematical Society.[1]Contents1 Selected publications 2 See also 3 References 4 External linksSelected publications[edit]Soare, R. (1987). Recursively enumerable sets and degrees. Perspectives in Mathematical Logic [...More...]  "Robert I. Soare" on: Wikipedia Yahoo Parouse 

Compact Space In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other). Examples include a closed interval, a rectangle, or a finite set of points. This notion is defined for more general topological spaces than Euclidean space Euclidean space in various ways. One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass theorem Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded [...More...]  "Compact Space" on: Wikipedia Yahoo Parouse 

Analytical Hierarchy In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of secondorder arithmetic, which can have quantifiers over both the set of natural numbers, N displaystyle mathbb N , and over functions from N displaystyle mathbb N to N displaystyle mathbb N [...More...]  "Analytical Hierarchy" on: Wikipedia Yahoo Parouse 

Arithmetical Comprehension In mathematical logic, secondorder arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. It was introduced by David Hilbert and Paul Bernays Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of secondorder arithmetic is denoted Z2. Secondorder arithmetic includes, but is significantly stronger than, its firstorder counterpart Peano arithmetic. Unlike Peano arithmetic, secondorder arithmetic allows quantification over sets of natural numbers as well as numbers themselves. Because real numbers can be represented as (infinite) sets of natural numbers in wellknown ways, and because second order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in secondorder arithmetic [...More...]  "Arithmetical Comprehension" on: Wikipedia Yahoo Parouse 

Zermelo–Fraenkel Set Theory In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Zermelo–Fraenkel set theory Zermelo–Fraenkel set theory with the historically controversial axiom of choice included is commonly abbreviated ZFC, where C stands for choice.[1] Many authors use ZF to refer to the axioms of Zermelo–Fraenkel set theory Zermelo–Fraenkel set theory with the axiom of choice excluded. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. ZFC is intended to formalize a single primitive notion, that of a hereditary wellfounded set, so that all entities in the universe of discourse are such sets [...More...]  "Zermelo–Fraenkel Set Theory" on: Wikipedia Yahoo Parouse 

Hartley Rogers, Jr. Hartley Rogers Jr. (1926–2015) was a mathematician who worked in recursion theory, and was a professor in the Mathematics Department of the Massachusetts Institute of Technology. The Rogers equivalence theorem is named after him. Born in 1926 in Buffalo, New York,[1] he studied under Alonzo Church at Princeton, and received his Ph.D. there in 1952. He served on the MIT faculty from 1956 until his death, July 17, 2015.[2] There he had been involved in many scholarly extracurricular activities, including running SPUR (Summer Program in Undergraduate Research) for MIT undergraduates, overseeing the mathematics section of RSI (Research Science Institute) for advanced high school students, and coaching the MIT Putnam exam team for nearly two decades starting in 1990, including the years 2003 and 2004 when MIT won for the first time since 1979 [...More...]  "Hartley Rogers, Jr." on: Wikipedia Yahoo Parouse 

Azriel Lévy Azriel Lévy[1] (Hebrew: עזריאל לוי; born c. 1934) is an Israeli mathematician, logician, and a professor emeritus at the Hebrew Hebrew University of Jerusalem.Contents1 Biography 2 Selected works 3 Notes 4 References 5 External linksBiography[edit] Lévy obtained his Ph.D. at the Hebrew Hebrew University of Jerusalem in 1958, under the supervision of Abraham Fraenkel Abraham Fraenkel and Abraham Robinson. Using Cohen's method of forcing, he proved several results on the consistency of various statements contradicting the axiom of choice. For example, with J. D. Halpern he proved that the Boolean prime ideal theorem does not imply the axiom of choice. He discovered the models L[x] used in inner model theory [...More...]  "Azriel Lévy" on: Wikipedia Yahoo Parouse 