HOME TheInfoList.com
Providing Lists of Related Topics to Help You Find Great Stuff
[::MainTopicLength::#1500] [::ListTopicLength::#1000] [::ListLength::#15] [::ListAdRepeat::#3] picture info 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). 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 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 Google 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 Effectively closed sets are a topic of study in classical computability theory [...More...] "Basis Theorem (computability)" on: Wikipedia Google Yahoo Parouse picture info Luitzen Egbertus Jan Brouwer Luitzen Egbertus Jan Brouwer ForMemRS (/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. 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 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 Google Yahoo Parouse picture info 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. Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises [...More...] "Theorem" on: Wikipedia Google 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 program-input 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 Google 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 Google 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^
.