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Diagonal Lemma
In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories. A particular instance of the diagonal lemma was used by Kurt Gödel in 1931 to construct his proof of the incompleteness theorems as well as in 1933 by Tarski to prove his undefinability theorem. In 1934, Carnap was the first to publish the diagonal lemma at some level of generality. The diagonal lemma is named in reference to Cantor's diagonal argument in set and number theory. The diagonal lemma applies to any sufficiently strong theories capable of representing the diagonal function. Such theories include first-order Peano arithmetic \mathsf, the weaker Robinson arithmetic \mathsf as well as any theory containing \mathsf (i.e. which interprets it). A common statement of the lemma (as given below) makes the stronger assumption that the theory can represent all recurs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Concept Of Truth In Formalized Languages
Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic". The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system. History In 1931, Kurt Gödel published the incompleteness theorems, which he proved in part by showing how to represent the syntax of formal logic within first-order arithmetic. Each expression of the formal language of arithmetic is assigned a distinct number. This procedure is known variously as Gödel numbering, ''coding'' and, more generally, as arithmetization. In particular, various ''sets'' of expressions are coded as sets of numbers. For various syntactic properties (such as ''being a formula'', ''being a sentence'', etc.), these s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Jeffrey
Richard Carl Jeffrey (August 5, 1926 – November 9, 2002) was an American philosopher, logician, and probability theorist. He is best known for developing and championing the philosophy of radical probabilism and the associated heuristic of probability kinematics, also known as Jeffrey conditioning. Life and career Born in Boston, Massachusetts, Jeffrey served in the U.S. Navy during World War II. As a graduate student he studied under Rudolf Carnap and Carl Hempel. He received his M.A. from the University of Chicago in 1952 and his Ph.D. from Princeton in 1957. After holding academic positions at MIT, City College of New York, Stanford University, and the University of Pennsylvania, he joined the faculty of Princeton in 1974 and became a professor emeritus there in 1999. He was also a visiting professor at the University of California, Irvine. Jeffrey, who died of lung cancer at the age of 76, was known for his sense of humor, which often came through in his breezy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Boolos
George Stephen Boolos (; September 4, 1940 – May 27, 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology. Life Boolos was of Greek-Jewish descent. He graduated with an A.B. in mathematics from Princeton University after completing a senior thesis, titled "A simple proof of Gödel's first incompleteness theorem", under the supervision of Raymond Smullyan. Oxford University awarded him the B.Phil. in 1963. In 1966, he obtained the first PhD in philosophy ever awarded by the Massachusetts Institute of Technology, under the direction of Hilary Putnam. After teaching three years at Columbia University, he returned to MIT in 1969, where he spent the rest of his career. A charismatic speaker well known for his clarity and wit, he once delivered a lecture (1994b) giving an account of Gödel's second incompleteness theorem, employing only words of one syllable. At the end of his viva, Hilary Putnam asked him, "A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-referential Paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their own article in this encyclopedia. These paradoxes may be due to fallacious reasoning ( falsidical), or an unintuitive solution ( veridical). The term ''paradox'' is often used to describe a counter-intuitive result. However, some of these paradoxes qualify to fit into the mainstream viewpoint of a paradox, which is a self-contradictory result gained even while properly applying accepted ways of reasoning. These paradoxes, often called ''antinomy,'' point out genuine problems in our understanding of the ideas of truth and description. Logic * : The supposition that, "if one of two simultaneous assumptions leads to a contradiction, the other assumption is also disproved" leads to paradoxical consequences. Not to be confused with the Bar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-reference
Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields. In natural or formal languages, self-reference occurs when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philosophy, self-reference also refers to the ability of a subject to speak of or refer to itself, that is, to have the kind of thought expressed by the first person nominative singular pronoun "I" in English. Self-reference is studied and has applications in mathematics, philosophy, computer programming, second-order cybernetics, and linguistics, as well as in humor. Self-referential statements are sometimes paradoxical, and can also be considered recursive. In logic, mathematics and computing In classical philosophy, paradoxes were created b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Fixed Point Theorems
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. In mathematical analysis The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed-point theorem (1911) is a non- constructive result: it says that any continuous function from the closed unit ball in ''n''-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (see also Sperner's lemma). For example, the cosine function is continuous in ��1, 1and maps it into ��1, 1 and thus must have a fixed point. This is clear when examining a sketched graph of the cosine function; the fixed point occurs where the cosine curve ''y'' = cos(''x'') intersects the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indirect Self-reference
Indirect self-reference describes an object referring to itself indirectly. For example, the "this sentence is false." contains a direct self-reference, in which the phrase "this sentence" refers directly to the sentence as a whole. An indirectly self-referential sentence would replace the phrase "this sentence" with an expression that effectively still referred to the sentence, but did not use the pronoun "this." If the quine of a phrase is defined to be the quotation of the phrase followed by the phrase itself, then the quine of: is a sentence fragment would be: "is a sentence fragment" is a sentence fragment which, incidentally, is a true statement. Now consider the sentence: "when quined, makes quite a statement" when quined, makes quite a statement The quotation here, plus the phrase "when quined," indirectly refers to the entire sentence. The importance of this fact is that the remainder of the sentence, the phrase "makes quite a statement," can now make a statement abou ... [...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 inspi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Löb's Theorem
In mathematical logic, Löb's theorem states that in Peano arithmetic (PA) (or any formal system including PA), for any formula ''P'', if it is provable in PA that "if ''P'' is provable in PA then ''P'' is true", then ''P'' is provable in PA. If Prov(''P'') is the assertion that the formula ''P'' is provable in PA, we may express this more formally as :If :\mathit \vdash :then :\mathit \vdash P. An immediate corollary (the contrapositive) of Löb's theorem is that, if ''P'' is not provable in PA, then "if ''P'' is provable in PA, then ''P'' is true" is not provable in PA. For example, "If 1+1=3 is provable in PA, then 1+1=3" is not provable in PA. Löb's theorem is named for Martin Hugo Löb, who formulated it in 1955. It is related to Curry's paradox. Löb's theorem in provability logic Provability logic abstracts away from the details of encodings used in Gödel's incompleteness theorems by expressing the provability of \phi in the given system in the language of modal logi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leon Henkin
Leon Albert Henkin (April 19, 1921, Brooklyn, New York – November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the Type theory, theory of types. He was an active scholar at the University of California, Berkeley, University of California, Berkeley, where he made great contributions as a researcher and teacher, as well as in administrative positions. At this university he directed, together with Alfred Tarski, the ''Group in Logic and the Methodology of Science'', from which many important logicians and philosophers emerged. He had a strong sense of social commitment and was a passionate defender of his pacifist and progressive ideas. He took part in many social projects aimed at teaching mathematics, as well as projects aimed at supporting women's and minority groups to pursue careers in mathematics and related fields. A lover of dance and literature, he appreciated life in all its facets: ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computability Theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since expanded to include the study of generalized computability and definable set, definability. In these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: * What does it mean for a function (mathematics), function on the natural numbers to be computable? * How can noncomputable functions be classified into a hierarchy based on their level of noncomputability? Although there is considerable overlap in terms of knowledge and methods, mathematical computability theorists study the theory of relative computability, reducibility notions, and degree structures; those in the computer science field focus on the theory of computational complexity theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
