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Paris–Harrington Theorem
In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory, namely the strengthened finite Ramsey theorem, is true, but not provable in Peano arithmetic. This has been described by some (such as the editor of the ''Handbook of Mathematical Logic'' in the references below) as the first "natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic; it was already known that such statements existed by Gödel's first incompleteness theorem. Strengthened finite Ramsey theorem The strengthened finite Ramsey theorem is a statement about colorings and natural numbers and states that: : For any positive integers ''n'', ''k'', ''m'', such that ''m ≥ n'', one can find ''N'' with the following property: if we color each of the ''n''-element subsets of ''S'' = with one of ''k'' colors, then we can find a subset ''Y'' of ''S'' with at least ''m'' elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel's Incompleteness Theorems
Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ... that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistency, consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kruskal's Tree Theorem
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding. History The theorem was conjectured by Andrew Vázsonyi and proved by ; a short proof was given by . It has since become a prominent example in reverse mathematics as a statement that cannot be proved within ATR0 (a form of arithmetical transfinite recursion), and a finitary application of the theorem gives the existence of the fast-growing TREE function. In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function. Statement The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree with a root, and given vertices , , call a successor of if the unique path from the root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kanamori–McAloon Theorem
In mathematical logic, the Kanamori–McAloon theorem, due to , gives an example of an incompleteness in Peano arithmetic, similar to that of the Paris–Harrington theorem. They showed that a certain finitistic theorem in Ramsey theory is not provable in Peano arithmetic (PA). Statement Given a set s\subseteq\mathbb of non-negative integers, let min(s) denote the minimum element of s. Let n denote the set of all ''n''-element subsets of X. A function f: n\rightarrow\mathbb where X\subseteq\mathbb is said to be ''regressive'' if f(s) |
Goodstein's Theorem
In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every ''Goodstein sequence'' eventually terminates at 0. Kirby and Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic). This was the third example of a true statement that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic. The Paris–Harrington theorem gave another example. Laurence Kirby and Jeff Paris introduced a graph-theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" (named for the mythological multi-headed Hydra of Lerna) is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ackermann Function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication of his function (which had three non-negative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version, the two-argument Ackermann–Péter function is defined as follows for nonnegative integers ''m'' and ''n'': : \begin \operatorname(0, n) & = & n + 1 \\ \operatorname(m+1, 0) & = & \operatorname(m, 1) \\ \operatorname(m+1, n+1) & = & \operatorname(m, \operatorname(m+1, n)) \end Its value grows rapidly, even for small inputs. For example, is an integer o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Recursive Function
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined before entering the loop). Primitive recursive functions form a strict subset of those general recursive functions that are also total functions. The importance of primitive recursive functions lies in the fact that most computable functions that are studied in number theory (and more generally in mathematics) are primitive recursive. For example, addition and division, the factorial and exponential function, and the function which returns the ''n''th prime are all primitive recursive. In fact, for showing that a computable function is primitive recursive, it suffices to show that its time complexity is bounded above by a primitive recursive function of the input size. It is hence not that easy to devise a computable function that is ''n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computable Function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output. Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the general recursive functions. Before the precise definition of computable function, mathematicians often used the informal term ''effectively calculable''. This term has since come to be identified with the com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leo Harrington
Leo Anthony Harrington (born May 17, 1946) is a professor of mathematics at the University of California, Berkeley who works in recursion theory, model theory, and set theory. Having retired from being a Mathematician, Professor Leo Harrington is now a Philosopher. His notable results include proving the Paris–Harrington theorem along with Jeff Paris, showing that if the axiom of determinacy holds for all analytic sets then ''x''# exists for all reals ''x'', and proving with Saharon Shelah that the first-order theory of the partially ordered set of recursively enumerable Turing degrees In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set. Overview The concept of Turing degree is fund ... is undecidable. References External linksHome page * Living people American logicians 20th-century American mathematicians 21st-cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Principle
In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle are often used for enumerative purposes. Bijective proofs are utilized to demonstrate that two sets have the same number of elements. The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a discrete context. Many combinatorial identities arise from double counting methods or the method of distinguished element. Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations. Rule of sum The rule of sum is an intuitive principle stating that if there are ''a'' possible outcomes for an event (or ways to do something) and ''b'' possible outcomes for another event (or ways to do another thin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeff Paris (mathematician)
Jeffrey Bruce Paris (; born 15 November 1944) is a British mathematician and Professor of Logic in the School of Mathematics, University of Manchester, School of Mathematics at the University of Manchester. Education Paris gained his doctorate supervised by Robin Gandy at Manchester in 1969 with a dissertation on ''Large Cardinals and the Generalized Continuum Hypothesis''. Research and career Paris is known for his work on mathematical logic, in particular provability logic, provability in arithmetic, uncertain reasoning and inductive logic with an emphasis on rationality and common sense principles. Awards and honours Paris was awarded the Whitehead Prize in 1983 and elected a Fellow of the British Academy (FBA) in 1999. Personal life Paris was married to Malvyn Loraine Blackburn until 1983 when he married Alena Vencovská. He has three sons and three daughters including runner Jasmin Paris. References 20th-century British mathematicians 21st-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-order Arithmetic
In mathematical logic, second-order 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. A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book ''Grundlagen der Mathematik''. The standard axiomatization of second-order arithmetic is denoted by Z2. Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order 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 well-known ways, and because second-order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, secon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
