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Berry's Paradox
The Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters" (a phrase with fifty-seven letters). Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G. Berry (1867–1928), a junior librarian at Oxford's Bodleian Library. Russell called Berry "the only person in Oxford who understood mathematical logic". The paradox was called "Richard's paradox" by Jean-Yves Girard. Overview Consider the expression: :"The smallest positive integer not definable in under sixty letters." Since there are only twenty-six letters in the English alphabet, there are finitely many phrases of under sixty letters, and hence finitely many positive integers that are defined by phrases of under sixty letters. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under sixty letters. If there are positive integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-referential
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 Cretans are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gregory Chaitin
Gregory John Chaitin ( ; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem. He is considered to be one of the founders of what is today known as algorithmic (Solomonoff–Kolmogorov–Chaitin, Kolmogorov or program-size) complexity together with Andrei Kolmogorov and Ray Solomonoff. Along with the works of e.g. Solomonoff, Kolmogorov, Martin-Löf, and Leonid Levin, algorithmic information theory became a foundational part of theoretical computer science, information theory, and mathematical logic. It is a common subject in several computer science curricula. Besides computer scientists, Chaitin's work draws attention of many philosophers and mathematicians to fundamental problems in mathematical creativity and digital philosophy. Mathematics and comp ... [...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 paradoxical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Paradoxes
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1925–1927, it appeared in a second edition with an important ''Introduction to the Second Edition'', an ''Appendix A'' that replaced ✸9 and all-new ''Appendix B'' and ''Appendix C''. ''PM'' is not to be confused with Russell's 1903 ''The Principles of Mathematics''. ''PM'' was originally conceived as a sequel volume to Russell's 1903 ''Principles'', but as ''PM'' states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was originally intended by us to be comprised in a second volume of ''Principles of Mathematics''... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred N
Alfred may refer to: Arts and entertainment *''Alfred J. Kwak'', Dutch-German-Japanese anime television series * ''Alfred'' (Arne opera), a 1740 masque by Thomas Arne * ''Alfred'' (Dvořák), an 1870 opera by Antonín Dvořák *"Alfred (Interlude)" and "Alfred (Outro)", songs by Eminem from the 2020 album ''Music to Be Murdered By'' Business and organisations * Alfred, a radio station in Shaftesbury, England *Alfred Music, an American music publisher *Alfred University, New York, U.S. *The Alfred Hospital, a hospital in Melbourne, Australia People * Alfred (name) includes a list of people and fictional characters called Alfred * Alfred the Great (848/49 – 899), or Alfred I, a king of the West Saxons and of the Anglo-Saxons Places Antarctica * Mount Alfred (Antarctica) Australia * Alfredtown, New South Wales * County of Alfred, South Australia Canada * Alfred and Plantagenet, Ontario * Alfred Island, Nunavut * Mount Alfred, British Columbia United States * Alfred, Maine, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bhartṛhari
Bhartṛhari (Devanagari: ; also romanised as Bhartrihari; fl. c. 5th century CE) was a Hindu linguistic philosopher to whom are normally ascribed two influential Sanskrit texts: * the ''Vākyapadīya'', on Sanskrit grammar and linguistic philosophy, a foundational text in the Indian grammatical tradition, explaining numerous theories on the word and on the sentence, including theories which came to be known under the name of Sphoṭa; in this work Bhartrhari also discussed logical problems such as the liar paradox and a paradox of unnameability or unsignfiability which has become known as Bhartrhari's paradox, and * the ''Śatakatraya'', a work of Sanskrit poetry, comprising three collections of about 100 stanzas each; it may or may not be by the same author who composed the two mentioned grammatical works. In the medieval tradition of Indian scholarship, it was assumed that both texts were written by the same person. Modern philologists were sceptical of this claim, owing t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turing Machines
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm. The machine operates on an infinite memory tape divided into discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write and which direction to move is based on a finite table that specifies what to do for each comb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or ''well-formed formulas''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with particular meanings or semantics. In computational complexity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Data Compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information. Typically, a device that performs data compression is referred to as an encoder, and one that performs the reversal of the process (decompression) as a decoder. The process of reducing the size of a data file is often referred to as data compression. In the context of data transmission, it is called source coding; encoding done at the source of the data before it is stored or transmitted. Source coding should not be confused with channel coding, for error detection and correction or line coding, the means for mapping data onto a signal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the non-linguistic bearer of truth or falsity which makes any sentence that expresses it either true or false. While the term "proposition" may sometimes be used in everyday language to refer to a linguistic statement which can be either true or false, the technical philosophical term, which differs from the mathematical usage, refers exclusively to the non-linguistic meaning behind the statement. The term is often used very broadly and can also refer to various related concepts, both in the history of philosophy and in contemporary analytic philosophy. It can generally be used to refer to some or all of the following: The primary bearers of truth values (such as "true" and "false"); the objects of belief and other propositional attitudes (i.e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
