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Gödel's incompleteness theorems

Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results 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, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). Background[edit] Many theories of interest include an infinite set of axioms, however. A formal theory is said to be effectively generated if its set of axioms is a recursively enumerable set. p ↔ F(G(p)). B.

Perth, Western Australia As part of Perth's role as the capital of Western Australia, the state's Parliament and Supreme Court are located within the city, as well as Government House, the residence of the Governor of Western Australia. Perth became known worldwide as the "City of Light" when city residents lit their house lights and streetlights as American astronaut John Glenn passed overhead while orbiting the earth on Friendship 7 in 1962.[10][11] The city repeated the act as Glenn passed overhead on the Space Shuttle in 1998.[12][13] Perth came 9th in the Economist Intelligence Unit's August 2012 list of the world's most liveable cities,[14] and was classified by the Globalization and World Cities Research Network in 2010 as a world city.[15] History[edit] Indigenous history[edit] The area where Perth now stands was called Boorloo by the Aborigines living there in 1827 at the time of their first contact with Europeans. Early European sightings[edit] Swan River Colony[edit] Federation and beyond[edit]

Theorem of the Day The Whole Jolly Lot (now enriched with The list is presented here in reverse chronological order, so that new additions will appear at the top. This is not the order in which the theorem of the day is picked which is more designed to mix up the different areas of mathematics and the level of abstractness or technicality involved. The way that the list of theorems is indexed is described here. Every theorem number is linked to its entry in the delightful 'Prime Curios!' All files are pdf , mostly between 100 and 300 Kbytes in size. A QED following a theorem indicates that the description includes a proof of the theorem. 211 Willans' Formula QED 210 The Basel Problem QED 209 The Erdős Discrepancy Conjecture QED ( a Theorem under construction!) 208 Toricelli's Trumpet QED 207 The Eratosthenes-Legendre Sieve QED 206 Euler's Formula QED 205 The Classification of the Semiregular Tilings 204 Singmaster's Binomial Multiplicity Bound QED ( 203 Euler's Continued Fraction Correspondence 191 L'Hospital's Rule

Linear function In mathematics, the term linear function refers to two different, although related, notions:[1] As a polynomial function[edit] In calculus, analytic geometry and related areas, a linear function is a polynomial of degree one or less, including the zero polynomial (the latter not being considered to have degree zero). For a function of any finite number independent variables, the general formula is and the graph is a hyperplane of dimension k. A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. In this context, the other meaning (a linear map) may be referred to as a homogeneous linear function or a linear form. As a linear map[edit] In linear algebra, a linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication: Some authors use "linear function" only for linear maps that take values in the scalar field;[4] these are also called linear functionals. See also[edit]

Ground expression Term that does not contain any variables is a ground formula, with and being constant symbols. Examples[edit] for the numbers 0 and 1, respectively, a unary function symbol for the successor function and a binary function symbol for addition. are ground terms; are ground terms; are ground terms; and are terms, but not ground terms; and are ground formulae. Formal definitions[edit] What follows is a formal definition for first-order languages. the set of constant symbols, the set of functional operators, and the set of predicate symbols. Ground term[edit] Elements of are ground terms;If is an -ary function symbol and are ground terms, then is a ground term.Every ground term can be given by a finite application of the above two rules (there are no other ground terms; in particular, predicates cannot be ground terms). Roughly speaking, the Herbrand universe is the set of all ground terms. Ground atom[edit] If is an -ary predicate symbol and are ground terms, then is a ground predicate or ground atom.

Gödel, Escher, Bach Gödel, Escher, Bach: An Eternal Golden Braid (pronounced [ˈɡøːdəl ˈɛʃɐ ˈbax]), also known as GEB, is a 1979 book by Douglas Hofstadter, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll".[1] By exploring common themes in the lives and works of logician Kurt Gödel, artist M. C. Escher and composer Johann Sebastian Bach, GEB expounds concepts fundamental to mathematics, symmetry, and intelligence. In response to confusion over the book's theme, Hofstadter has emphasized that GEB is not about mathematics, art, and music but rather about how cognition and thinking emerge from well-hidden neurological mechanisms. Structure[edit] GEB takes the form of an interweaving of various narratives. One dialogue in the book is written in the form of a crab canon, in which every line before the midpoint corresponds to an identical line past the midpoint. Themes[edit] Puzzles[edit] The book is filled with puzzles. Impact[edit] Translation[edit]

Hammack Home This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative. Also see the Mathematical Association of America Math DL review (of the 1st edition), and the Amazon reviews. The second edition is identical to the first edition, except some mistakes have been corrected, new exercises have been added, and Chapter 13 has been extended. Order a copy from Amazon or Barnes & Noble for $13.75 or download a pdf for free here. Part I: Fundamentals Part II: How to Prove Conditional Statements Part III: More on Proof Part IV: Relations, Functions and Cardinality Thanks to readers around the world who wrote to report mistakes and typos! Instructors: Click here for my page for VCU's MATH 300, a course based on this book. I will always offer the book for free on my web page, and for the lowest possible price through on-demand publishing.

Polynomial The graph of a polynomial function of degree 3 Etymology[edit] According to the Oxford English Dictionary, polynomial succeeded the term binomial, and was made simply by replacing the Latin root bi- with the Greek poly-, which comes from the Greek word for many. The word polynomial was first used in the 17th century.[1] Notation and terminology[edit] It is a common convention to use upper case letters for the indeterminates and the corresponding lower case letters for the variables (arguments) of the associated function. It may be confusing that a polynomial P in the indeterminate X may appear in the formulas either as P or as P(X). Normally, the name of the polynomial is P, not P(X). In particular, if a = X, then the definition of P(a) implies This equality allows writing "let P(X) be a polynomial" as a shorthand for "let P be a polynomial in the indeterminate X". Definition[edit] A polynomial in a single indeterminate can be written in the form where For example: is a term. then is from

Grundlagen der Mathematik From Wikipedia, the free encyclopedia Two-volume work by David Hilbert and Paul Bernays 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. Publication history[edit] 1934/1939 (Vol. See also[edit] Hilbert–Bernays paradox References[edit] External links[edit] Hilbert Bernays Project which aims to produce an English translation of Grundlagen der Mathematik.

Principe de relativité Un article de Wikipédia, l'encyclopédie libre. Le principe de relativité[1] affirme que les lois physiques s'expriment de manière identique dans tous les référentiels inertiels. D'une théorie à l'autre (physique classique, relativité restreinte ou générale), la formulation du principe a évolué et s'accompagne d'autres hypothèses sur l'espace et le temps, sur les vitesses, etc. Certaines de ces hypothèses étaient implicites ou « évidentes » en physique classique, car conformes à toutes les expériences, et elles sont devenues explicites et plus discutées à partir du moment où la relativité restreinte a été formulée. Exemples en physique classique[modifier | modifier le code] Première situation Supposons que dans un train roulant à vitesse constante (sans les accélérations, petites ou grandes, perceptibles dans le cas d'un train réel), un voyageur se tient debout, immobile par rapport à ce train, et tient un objet dans la main. Deuxième situation Conclusion Propriété : soit ( ), alors ( ) et ( ) et

Cantor and Cohen: Infinite investigators part I June 2008 This is one half of a two-part article telling a story of two mathematical problems and two men: Georg Cantor, who discovered the strange world that these problems inhabit, and Paul Cohen (who died last year), who eventually solved them. The first of these problems — the axiom of choice — is the subject of this article, while the other article explores what is known as the continuum hypothesis. Each article is self-contained, so you don't have to read both to get the picture. Cantor: The infinite match-maker Georg Cantor was a German logician who, in the late 19th century, achieved a feat which scientists, philosophers, and theologians had previously only dreamed about: a detailed analysis of infinity. Georg Cantor Cantor's discovery was that there is not just one infinity, but a never-ending hierarchy, each infinitely bigger than the last. Suppose you have two collections of objects. Sets and politics Paradoxes and axioms But others were more receptive to Cantor's ideas. , written