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Subformula Property
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics". of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applicati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory. Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provability Logic
Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic. Examples There are a number of provability logics, some of which are covered in the literature mentioned in . The basic system is generally referred to as GL (for Gödel– Löb) or L or K4W (W stands for well-foundedness). It can be obtained by adding the modal version of Löb's theorem to the logic K (or K4). Namely, the axioms of GL are all tautologies of classical propositional logic plus all formulas of one of the following forms: * Distribution axiom: * Löb's axiom: And the rules of inference are: * ''Modus ponens'': From ''p'' → ''q'' and ''p'' conclude ''q''; * Necessitation: From \vdash ''p'' conclude \vdash . History The GL model was pioneered by Robert M. Solovay in 1976. Since then, until his death in 1996, the prime inspirer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetical Hierarchy
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy is important in recursion theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic. The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines. The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets. The arithmetical hierarchy of formulas The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted \Sigma^0_n and \Pi^0_n for natural numbers ''n'' (inclu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitary
In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values. In standard mathematics, an operation is finitary by definition. Therefore these terms are usually only used in the context of infinitary logic. Finitary argument A finitary argument is one which can be translated into a finite set of symbolic propositions starting from a finiteThe number of axioms ''referenced'' in the argument will necessarily be finite since the proof is finite, but the number of axioms from which these are ''chosen'' is infinite when the system has axiom schemes, e.g. the axiom schemes of propositional calculus. set of axioms. In other words, it is a proof (including all assumptions) that can be written on a large enough sheet of paper. By contrast, infinitary logic studies logics that allow infinitely long statements and proofs. In such a logic, one ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grundlagen Der Mathematik
''Grundlagen der Mathematik'' (English: ''Foundations of Mathematics'') is a two-volume work by David Hilbert and Paul Bernays. Originally published in 1934 and 1939, it presents fundamental mathematical ideas and introduced 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 precurs .... Publication history *1934/1939 (Vol. I, II) First German edition, Springer *1944 Reprint of first edition by J. W. Edwards, Ann Arbor, Michigan. *1968/1970 (Vol. I, II) Second revised German edition, Springer *1979/1982 (Vol. I, II) Russian translation of 1968/1970, Nauka Publ., Moscow *2001/2003 (Vol. I, II) French translation, L’Harmattan, Paris *2011/2013 (Parts A and B of Vol. I, prefaces and sections 1-5) English translation of 1968 and 1934, bilingual with German facsimile on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics. In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic. Life Early life and edu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Dedekind
Julius Wilhelm Richard Dedekind (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 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''. Life 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 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. He first attended the Collegium Carolinum in 1848 before transferring to the University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, artificial intelligence, cognitive science, computer science and various areas of analytic philosophy, especially philosophy of mathematics, philosophy of language, epistemology, and metaphysics.Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003. He was one of the early 20th century's most prominent logicians, and a founder of analytic philosophy, along with his predecessor Gottlob Frege, his friend and colleague G. E. Moore and his student and protégé Ludwig Wittgenstein. Russell with Moore led the British "revolt against idealism". Together with his former teacher A. N. Whitehead, Russell wrote ''Principia Mathematica'', a milestone in the development of classical logic, and a major attempt to reduce the whole ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern rigorous and systematic treatment of the method of mathematical induction. He spent most of his career teaching mathematics at the University of Turin. He also wrote an international auxiliary language, Latino sine flexione ("Latin without inflections"), which is a simplified version of Classical Latin. Most of his books and papers are in Latino sine flexione, others are in Italian. Biography Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Piedmont, Italy. He attended the Liceo classico Cavour in Turin, and enrolled at the University of Turin in 1876, graduatin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever. His contributions include the development of modern logic in the ''Begriffsschrift'' and work in the foundations of mathematics. His book the ''Foundations of Arithmetic'' is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof Complexity
In logic and theoretical computer science, and specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse the computational resources that are required to prove or refute statements. Research in proof complexity is predominantly concerned with proving proof-length lower and upper bounds in various propositional proof systems. For example, among the major challenges of proof complexity is showing that the Frege system, the usual propositional calculus, does not admit polynomial-size proofs of all tautologies. Here the size of the proof is simply the number of symbols in it, and a proof is said to be of polynomial size if it is polynomial in the size of the tautology it proves. Systematic study of proof complexity began with the work of Stephen Cook and Robert Reckhow (1979) who provided the basic definition of a propositional proof system from the perspective of computational complexity. Specifically Cook and Reckhow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
