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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 of the natural numbers—specifically those theories that are strong enough to represent all computable functions. The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's undefinability theorem. Background Let \mathbb be the set of natural numbers. A first-order theory T in the language of arithmetic ''represents'' the computable function f: \mathbb\rightarrow\mathbb if there exists a "graph" predicate \mathcal_f(x, y) in the language of T such that for each n \in \mathbb : \vdash_\,(\forall y) ^\circ f(n)=y) \Leftrightarrow \mathcal_f(^\circ n,\,y)/math> Here ^\circ n is the numeral corresponding to the natural number n, which is defined to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor's Diagonal Argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. English translation: Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began. The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874. However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the '' Entscheidungsproblem''. Diagonalization arguments are often also the source of contradictions like Russell's paradox and Richard's paradox ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,For instance, in their "Principia Mathematica' (''Stanford Encyclopedia of Philosophy'' edition). Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics, building on earlier work by the likes of Richard Dedekind, Georg Cantor and Frege. Gödel published his first incompleteness theorem in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any ω-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Boolos
George Stephen Boolos (; 4 September 1940 – 27 May 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology. Life Boolos is 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, ... [...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. Although considered paradoxes, some of these are simply based on fallacious reasoning ( falsidical), or an unintuitive solution ( veridical). Informally, the term ''paradox'' is often used to describe a counter-intuitive result. However, some of these paradoxes qualify to fit into the mainstream perception 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 paradoxi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-reference
Self-reference occurs in natural or formal languages 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, it 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 by self-referential concepts such as the omnipotence paradox of asking if it was possible for a being to exist so powerful that it could create a stone that it could not lift. The Epimenides paradox, 'All Cretan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Recursive Arithmetic
Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician , reprinted in translation in as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε0, which is the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called Skolem arithmetic. The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursive function, including the operations of addition, multiplication, and exponentiation. PRA cannot explicitly quantify over the domain of natural numbers. PRA is often taken as the basic metamathemati ... [...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. Some authors claim that results of this kind are amongst the most generally useful in mathematics. 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 cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indirect Self-reference
Indirect self-reference describes an object referring to itself ''indirectly''. For example, define the function f such that f(x) = x(x). Any function passed as an argument to f is invoked with itself as an argument, and thus in any use of that argument is indirectly referring to itself. This example is similar to the Scheme expression "((lambda(x)(x x)) (lambda(x)(x x)))", which is expanded to itself by beta reduction, and so its evaluation loops indefinitely despite the lack of explicit looping constructs. An equivalent example can be formulated in lambda calculus. Indirect self-reference is special in that its self-referential quality is not explicit, as it is in the sentence "this sentence is false." 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." An example will help ... [...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 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 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 subrecursive hierarchies, formal methods, and formal languages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene's Recursion Theorem
In computability theory, Kleene's recursion theorems are a pair of fundamental results about the application of computable functions to their own descriptions. The theorems were first proved by Stephen Kleene in 1938 and appear in his 1952 book ''Introduction to Metamathematics''. A related theorem, which constructs fixed points of a computable function, is known as Rogers's theorem and is due to Hartley Rogers, Jr. The recursion theorems can be applied to construct fixed points of certain operations on computable functions, to generate quines, and to construct functions defined via recursive definitions. Notation The statement of the theorems refers to an admissible numbering \varphi of the partial recursive functions, such that the function corresponding to index e is \varphi_e. If F and G are partial functions on the natural numbers, the notation F \simeq G indicates that, for each ''n'', either F(n) and G(n) are both defined and are equal, or else F(n) and G(n) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliott Mendelson
Elliott Mendelson (May 24, 1931 – May 7, 2020) was an American logician. He was a professor of mathematics at Queens College of the City University of New York, and the Graduate Center, CUNY. He was Jr. Fellow, Society of Fellows, Harvard University, 1956–58. Career Mendelson earned his BA from Columbia University and PhD from Cornell University. Mendelson taught mathematics at the college level for more than 30 years, and is the author of books on logic, philosophy of mathematics, calculus, game theory and mathematical analysis. His ''Introduction to Mathematical Logic'', first published in 1964, was reviewed by Dirk van Dalen who noted that it included "a large variety of subjects that should be part of the education of any mathematics student with an interest in foundational matters." Dirk van Dalen (1969Review: Introduction to Mathematical Logic Journal of Symbolic Logic 34(1): 110,1 via JSTOR Books Sole author * * * * * * * Co-author * * * * Editor * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudolf Carnap
Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. He is considered "one of the giants among twentieth-century philosophers." Biography Carnap's father had risen from being a poor ribbon-weaver to be the owner of a ribbon-making factory. His mother came from an academic family; her father was an educational reformer and her oldest brother was the archaeologist Wilhelm Dörpfeld. As a ten-year-old, Carnap accompanied Wilhelm Dörpfeld on an expedition to Greece. Carnap was raised in a profoundly religious Protestant family, but later became an atheist. He began his formal education at the Barmen Gymnasium and the Gymnasium in Jena. From 1910 to 1914, he attended the University of Jena, intending to write a thesis in physics. He also intently studied Immanuel Kant's '' Critique of Pu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
