pistons cams engine cycles mechanics materials might used choose appropriate materials frame engine fluid mechanics might used design ventilation system vehicle see hvac design intake system engine mechatronics robotics scorbot er u workbench cnc mill cnc lathe mechatronics combination mechanics electronics interdisciplinary branch mechanical engineering electrical engineering software engineering concerned integrating electrical mechanical engineering create hybrid automation systems way machines


automated use electric motors servo mechanisms electrical systems conjunction special software common example mechatronics system cd rom drive mechanical systems open close drive spin cd move laser optical system reads data cd converts bits integrated software controls process communicates contents cd computer robotics application mechatronics create robots often used industry perform tasks dangerous unpleasant repetitive robots may

Applied science
Engineering and technology
Mechanical Engineering


shape size preprogrammed interact physically world create robot engineer typically employs kinematics determine robot 's range motion mechanics determine stresses within robot robots used extensively industrial automation engineering allow businesses save money labor perform tasks either dangerous precise humans perform economically ensure better quality many companies employ assembly lines robots especially automotive industries factories robotized run


outside factory robots employed bomb disposal space exploration many fields robots also sold various residential applications recreation domestic applications structural analysis structural analysis branch mechanical engineering also civil engineering devoted examining objects fail fix objects performance structural failures occur two general modes static failure fatigue failure ''static structural failure'' occurs upon loaded force applied object analyzed either breaks deformed plastically depending criterion failure ''fatigue


employed determine cause structural analysis often used mechanical engineers failure occurred designing prevent failure engineers often use online documents books published asm aid determining type failure possible causes theory applied mechanical design physical testing often performed verify calculated results structural analysis may used office designing parts field analyze failed parts laboratories parts might undergo controlled failure tests thermodynamics thermo science thermodynamics


applied science used several branches engineering including mechanical chemical engineering simplest thermodynamics study energy use transformation system typically engineering thermodynamics concerned changing energy one form another example automotive engines convert chemical energy enthalpy fuel heat mechanical work eventually turns wheels thermodynamics principles used mechanical engineers fields heat transfer thermofluids energy conversion mechanical engineers use thermo science design engines power plants heating ventilation air conditioning hvac systems


heat exchangers heat sinks radiators refrigeration insulation others design drafting mechanical double seal drafting technical drawing means mechanical engineers design products create instructions manufacturing parts technical drawing computer model hand drawn schematic showing dimensions necessary manufacture part well assembly notes list required materials pertinent information us mechanical engineer skilled worker creates technical drawings may referred drafter draftsman drafting historically two dimensional process computer aided design cad programs


allow designer create three dimensions instructions manufacturing part must fed necessary machinery either manually programmed instructions use computer aided manufacturing cam combined cad cam program optionally engineer may also manually manufacture part using technical drawings however advent computer numerically controlled cnc manufacturing parts fabricated without need constant technician input manually manufactured parts generally consist spray coatings surface finishes processes cannot economically practically done machine drafting used


nearly every subdiscipline mechanical engineering many branches engineering architecture three dimensional models created using cad software also commonly used finite element analysis fea computational fluid dynamics cfd modern tools oblique view four cylinder inline crankshaft pistons many mechanical engineering companies especially industrialized nations begun incorporate computer aided engineering cae programs existing design analysis processes including solid modeling computer aided design cad method many benefits including easier exhaustive visualization products ability create virtual assemblies


parts ease use designing mating interfaces tolerances cae programs commonly used mechanical engineers include product lifecycle management plm tools analysis tools used perform complex simulations analysis tools may used predict product response expected loads including fatigue life manufacturability tools include finite element analysis fea computational fluid dynamics cfd computer aided manufacturing cam using cae programs mechanical design team quickly cheaply iterate design process develop product better meets cost performance constraints physical prototype need created


design nears completion allowing hundreds thousands designs evaluated instead relative addition cae analysis programs model complicated physical phenomena cannot solved hand viscoelasticity complex contact mating parts non newtonian flows mechanical engineering begins merge disciplines seen mechatronics multidisciplinary design optimization mdo used cae programs automate improve iterative design process mdo tools wrap around existing cae processes allowing product evaluation continue even analyst goes home day


also utilize sophisticated optimization algorithms intelligently explore possible designs often finding better innovative solutions difficult multidisciplinary design problems areas research mechanical engineers constantly pushing boundaries physically possible order produce safer cheaper efficient machines mechanical systems technologies cutting edge mechanical engineering listed see also exploratory engineering micro electro mechanical systems mems micron scale mechanical components springs gears fluidic heat transfer devices fabricated variety substrate materials silicon glass polymers like su examples mems


components accelerometers used car airbag sensors modern cell phones gyroscopes precise positioning microfluidic devices used biomedical applications friction stir welding fsw friction stir welding new type welding discovered welding institute twi innovative steady state non fusion welding technique joins materials previously un weldable including several aluminum alloys plays important role future construction airplanes potentially replacing rivets current uses technology date include welding seams aluminum main space shuttle external tank orion crew vehicle boeing delta ii


delta iv expendable launch vehicles spacex falcon rocket armor plating amphibious assault ships welding wings fuselage panels new eclipse aircraft eclipse aviation among increasingly growing pool uses composites composite cloth consisting woven carbon fiber composites composite materials combination materials provide different physical characteristics either material separately composite material research within mechanical engineering typically focuses designing subsequently finding applications stronger rigid materials attempting reduce weight susceptibility corrosion undesirable factors carbon


fiber reinforced composites instance used diverse applications spacecraft fishing rods mechatronics mechatronics synergistic combination mechanical engineering electronic engineering software engineering discipline mechatronics began way combine mechanical principles electrical engineering mechatronic concepts used majority electro mechanical systems typical electro mechanical sensors used mechatronics strain gauges thermocouples pressure transducers nanotechnology smallest scales mechanical engineering becomes nanotechnology one speculative goal create molecular assembler build molecules materials via mechanosynthesis goal remains


within exploratory engineering areas current mechanical engineering research nanotechnology include nanofilters nanofilms nanostructures among others finite element analysis finite element analysis computational tool used estimate stress strain deflection solid bodies uses mesh setup user defined sizes measure physical quantities node nodes higher precision field new basis finite element analysis fea finite element method fem dates back evolution computers made fea fem viable option analysis structural problems


many commercial codes nastran ansys abaqus widely used industry research design components modeling cad software packages added fea modules recent times cloud simulation platforms like simscale becoming common techniques finite difference method fdm finite volume method fvm employed solve problems relating heat mass transfer fluid flows fluid surface interaction etc biomechanics biomechanics application mechanical principles biological systems humans animals plants organs cells biomechanics also aids creating prosthetic limbs


artificial organs humans biomechanics closely related engineering often uses traditional engineering sciences analyze biological systems simple applications newtonian mechanics materials sciences supply correct approximations mechanics many biological systems past decade reverse engineering materials found nature bone matter gained funding academia structure bone matter optimized purpose bearing large amount compressive stress per unit weight goal replace crude steel bio material structural design past decade finite


mechanical engineering application acoustics acoustical engineering study sound vibration engineers work effectively reduce noise pollution mechanical devices buildings soundproofing removing sources unwanted noise study acoustics range designing efficient hearing aid microphone headphone recording studio enhancing sound quality orchestra hall acoustical engineering also deals vibration different mechanical systems related fields manufacturing engineering aerospace engineering automotive engineering grouped mechanical engineering times bachelor 's degree areas


typically difference specialized classes see also lists glossary mechanical engineering list historic mechanical engineering landmarks list inventors list mechanical engineering topics list mechanical engineers list related journals list mechanical electrical electronic equipment manufacturing companies revenue associations american society heating refrigerating air conditioning engineers ashrae american society mechanical engineers asme pi tau sigma mechanical engineering honor society society automotive engineers sae society women engineers swe institution mechanical engineers imeche british chartered institution building services engineers cibse british verein deutscher ingenieure vdi


''' media studies''' discipline field study deals content history effects various media particular mass media media studies may draw traditions social sciences humanities mostly core disciplines mass communication communication communication sciences communication studies researchers may also develop employ theories methods disciplines including cultural studies rhetoric including digital rhetoric philosophy literary theory psychology political science political economy economics sociology anthropology social theory art history criticism film theory information theory origin

Journalism, media studies and communication


first media studies program us introduced john culkin new school since graduated students culkin responsible bringing marshall mcluhan fordham subsequently founded center understanding media became new school program australia media studied broad subject states australia media studies australia first developed area study victorian universities early secondary schools mid today almost australian universities teach


media studies according government australia 's excellence research australia report leading universities country media studies ranked well world standards report 's scoring methodology monash university qut rmit university melbourne university queensland uts secondary schools early film studies course first began taught part victorian junior secondary curriculum mid early expanded media studies course taught course became part senior secondary curriculum later known victorian certificate


media teaching body known atom publishes ''metro'' ''screen education'' magazines canada canada media studies communication studies incorporated departments cover wide range approaches critical theory organizations research creation political economy example time research developed employ theories methods cultural studies philosophy political economy gender sexuality race theory management rhetoric film theory sociology anthropology harold innis marshall mcluhan famous canadian scholars contributions fields media ecology political economy th century


media studies media arts photography programs also originated programs started communication studies department concordia created late ryerson 's radio television film media photography programs renowned mid programs copied colleges universities nationally internationally today universities offer undergraduate degrees media communication studies many canadian scholars actively contribute field among brian massumi philosophy cultural studies kim sawchuk cultural studies feminist ageing studies carrie rentschler feminist theory fran ois cooren organizational communication


book understanding media extensions man media theorist marshall mcluhan suggested medium message human artefacts technologies media book introduced usage terms media language along precepts among global village age information medium anything mediates interaction world humans given perspective media study restricted media communications forms technology media users form ecosystem study ecosystem known


content writing speech written word content print print content telegraph change medium technology introduces human affairs message electric light used friday night football light desk could argue content electric light activities fact medium shapes controls form human association action makes message electric light looked communication medium n't


content electric light used spell brand name recognized medium similar radio mass media electric light eliminates time space factors human association creating deeper involvement mcluhan compared content juicy piece meat carried burglar distract watchdog mind effect medium made strong given another media content content movie book play maybe even opera ref name mcgraw hill ref mcluhan talks media


hot cold touches principle distinguishes one another hot medium ie radio movie extends single sense high definition high definition means state well filled data cool medium ie telephone tv considered low definition small amount data information given filled hot media low participation cool media high participation hot media low participation giving information excludes cool media high participation


speak simple attention grabbing terms search viewers leads emphasis sensational spectacular people complex nuanced views allowed hearing germany germany two main branches media theory media studies identified first major branch media theory roots humanities cultural studies film studies filmwissenschaft theater studies theaterwissenschaft german language literature studies germanistik well comparative literature studies komparatistik branch broadened substantially since


initial basis culturally based media studies often emphasised recently disciplinary title ''medienkulturwissenschaft'' germany primarily developed established plurality perspectives make difficult single one particular site branch medienwissenschaft originated frankfurt based theatre scholar hans theis lehmanns term post dramatic theater points directly increased blending co presence mediatized material german theater elsewhere since field theater studies onwards freie universit berlin led particular erika fischer lichte showed particular


interest ways theatricality influenced notions performativity aesthetic events within field film studies frankfurt berlin dominant development new perspectives moving image media heide schl pman frankfurt gertrud koch first bochum berlin key theorists contributing aesthetic theory cinema schl pmann ' 'dispositif'' moving image medium particularly context illusion koch many scholars became known media scholars germany originally scholars german friedrich kittler taught


humboldt universit zu berlin completed dissertation habilitation context ''germanistik'' one early publications new direction volume edited helmut kreuzer ''literature studies media studies'' ''literaturwissenschaft medienwissenschaft'' summarizes presentations given sseldorfer germanistentag second branch media studies germany comparable communication studies pioneered elisabeth noelle neumann branch studies mass media institutions effects society individuals german institute media communication policy founded media scholar lutz hachmeister


course content universities maintain practical journalistic training aim academic studies offer india media studies fast growing academic field india several dedicated departments research institutes view making best use communication facilities information publicity development government india sought advice ford foundation unesco team internationally known mass communication specialists recommended setting national institute training teaching research mass communication anna university first university start


master science electronic media programmes offers five year integrated programme two year programme electronic media department media sciences started january branching ugc 's educational multimedia research centre emmrc national institute open schooling world 's largest open schooling system offers mass communication subject studies senior secondary level major universities country mass media journalism studies departments centre study developing societies csds delhi media studies one major emphasis netherlands


netherlands media studies split several academic courses applied communication sciences communication information sciences communication media media culture theater film television sciences whereas communication sciences focuses way people communicate mediated unmediated media studies tends narrow communication mediated communication however would mistake consider media studies specialism communication sciences since media make small portion overall course indeed studies tend borrow elements one another communication sciences derivative thereof


studied erasmus university rotterdam radboud university tilburg university university amsterdam university groningen university twente roosevelt academy university utrecht vu university amsterdam wageningen university research centre media studies something similar studied university amsterdam vu university amsterdam erasmus university rotterdam university groningen university utrecht eight dutch universities collaborate overarching netherlands research school media studies rmes acts platform graduate students media research new zealand media studies new zealand healthy especially due renewed


activity country 's film industry taught secondary tertiary education institutes media studies nz regarded singular success subject well established tertiary sector http wwwwaikatoacnz film screen media studies university waikato media studies victoria university wellington film television media studies university auckland media studies massey university communication studies university otago different courses offer students range specialisations cultural studies media theory analysis practical film making journalism communications studies makes case


new zealand particularly significant respect media studies decade nationally mandated popular subject secondary high schools taught across three years structured developmental fashion scholarship media studies available academically gifted students according new zealand ministry education subject enrolment figures new zealand schools offered media studies subject representing students pakistan pakistan media studies programs widely offered university punjab lahore oldest department later


university karachi peshawar university bzu multaan islamia university bahwalpur also started communication programs newly established universities also offering mass communication program university gujrat emerged leading department bahria university established pakistan navy also offering bs media studies switzerland switzerland media communication studies offered several higher education institutions including international university geneva zurich university applied sciences university lugano university fribourg others united kingdom united kingdom media studies developed


head university 's centre mass communication research founder international association media communication research ref name mosco ref media studies taught uk taught key stages entry level gcse level scottish qualifications authority offers formal qualifications number different levels offered large area exam boards including aqa wjec much research field news media studies led reuters institute study journalism details research


projects results published risj annual report united states mass communication communication studies simply 'communication' names used far frequently media studies academic departments united states however focus programs sometimes excludes certain media film book publishing video games etc title media studies may used alone designate film studies rhetorical critical theory may appear combinations like media studies communication join two fields emphasize different focus involves study many emerging contemporary media


platforms social media boomed recent years broadcast cable tv longer primary form entertainment various screens offering worldwide events pastimes around clock proceeding examples evolution institutions reside within united states taken media studies wing grown mit comparative media studies program started leadership henry jenkins since growing graduate program mit 's largest humanities major following merger writing humanistic studies program roster


twenty faculty including pulitzer prize winning author junot diaz science fiction writer joe haldeman games scholar l taylor media scholars william uricchio cms co founder edward schiappa heather hendershot ref name ref named comparative media studies writing department places emphasis jenkins colleagues termed applied humanities hosts several research groups civic media digital humanities games computational media documentary mobile design groups used provide graduate students research assistantships cover cost tuition living expenses ref name incorporation writing humanistic studies also


placed mit 's science writing program writing across curriculum writing communications center roof around time mit interdisciplinary major university virginia department media studies officially established rapidly grown doubled size partly thanks acquisition professor siva vaidhyanathan cultural historian media scholar well inaugural verklin media policy ethics conference endowed ceo canoe ventures uva alumnus david verklin university california irvine professor mark poster


one first foremost theorists media culture us boasted strong department film media studies university california berkeley three institutional structures within media studies take place department film media formerly film studies program including famous theorists mary ann doane linda williams center new media long established interdisciplinary program formerly titled mass communications recently changed name media studies change eliminated connotations may accompany term mass former title recently


radford university virginia used title media studies department taught practitioner oriented major concentrations journalism advertising broadcast production web design programs combined previous department communication speech public relations create school communication media studies major radford still means someone concentrating journalism broadcasting advertising web production brooklyn college collaborated city university new york offer graduate studies television media since currently department television radio administers ms media studies hosts


center study world television see also anthropology media fan studies innis 's time space bias journalism market loyalties theory mass media mass communication mcluhan 's tetrad media effects media culture media echo chamber media ecology media literacy media psychology media system dependency mediatization media narcotizing dysfunction social aspects television sociology structural transformation public sphere transparency humanities uses gratifications theory references external links https webarchiveorg web http wwwdoingmediastudiescom media studies text production context' media studies paul long tim wall birmingham city university https webarchiveorg web http gdstrudley webscom media literacy


primary school children far primary school children knowledge skills access media make sense representations images produced create grant strudley university reading' humanities


'''mechanics''' greek area mathematics physics concerned motions physical objects specifically relationships among force matter motion forces applied objects result displacements changes object 's position relative environment branch physics origins ancient greece writings aristotle archimedes see history classical mechanics timeline classical mechanics early modern period scientists galileo kepler newton laid foundation known classical mechanics branch classical


physics deals particles either rest moving velocities significantly less speed light also defined branch science deals motion forces bodies quantum realm history antiquity main theory mechanics antiquity aristotelian mechanics later developer tradition hipparchus ref name mechanics http golemphutexasedu category tiny taste history ofhtml tiny taste history mechanics university texas austin ref medieval age arabic machine manuscript unknown date guess th th centuries musical toy


'force' 'inclination' called mayl argued object gained mayl object opposition natural motion concluded continuation motion attributed inclination transferred object object motion mayl spent also claimed projectile vacuum would stop unless acted upon conception motion consistent newton 's first law motion inertia states object motion stay motion unless acted


fundamental dynamic law namely constant force produces uniform motion thus anticipation vague fashion fundamental law classical mechanics namely force applied continuously produces acceleration influenced earlier writers ibn sina ref name ibn sina buridan al baghdaadi ref name gutman ref th century french priest jean buridan developed theory impetus later developed modern theories inertia velocity acceleration momentum work others developed th century england oxford calculators thomas bradwardine studied formulated various


laws regarding falling bodies concept main properties body uniformly accelerated motion falling bodies worked th century oxford calculators early modern age first european depiction piston pump taccola c two central figures early modern age galileo galilei isaac newton galileo 's final statement mechanics particularly falling bodies ''two new sciences'' newton 's ''philosophi naturalis principia mathematica'' provided detailed mathematical account mechanics using newly developed mathematics calculus providing basis newtonian mechanics ref


name mechanics th century windmill mechanics ireland dispute priority various ideas newton 's ''principia'' certainly seminal work tremendously influential systematic mathematics therein could stated earlier calculus developed however many ideas particularly pertain inertia impetus falling bodies developed stated earlier researchers recent galileo less known medieval predecessors precise credit times difficult contentious scientific language standards proof changed whether medieval statements ''equivalent''


modern statements ' 'sufficient'' proof instead ' 'similar'' modern statements ''hypotheses'' often debatable modern age two main modern developments mechanics general relativity einstein quantum mechanics developed th century based part earlier th century ideas development modern continuum mechanics particularly areas elasticity plasticity fluid dynamics electrodynamics thermodynamics deformable media started second half th century types mechanical bodies often used term '''body''' needs stand wide assortment objects including particles projectiles spacecraft stars parts


machinery parts solids parts fluids gases liquids etc distinctions various sub disciplines mechanics concern nature bodies described particles bodies little known internal structure treated mathematical points classical mechanics rigid bodies size shape retain simplicity close particle adding called degrees freedom orientation space otherwise bodies may semi rigid ie elastic non rigid ie fluid subjects classical quantum divisions study instance motion spacecraft


gravitational particles generate fields acting sources fact quantum mechanics particles fields described theoretically wave function classical prof walter lewin explains newton 's law gravitation mit course following described forming classical mechanics newtonian mechanics original theory motion kinematics forces dynamics analytical mechanics reformulation newtonian mechanics emphasis system energy rather forces two main branches analytical mechanics hamiltonian mechanics theoretical formalism based principle conservation energy lagrangian mechanics


another theoretical formalism based principle least action classical statistical mechanics generalizes ordinary classical mechanics consider systems unknown state often used derive thermodynamic properties celestial mechanics motion bodies space planets comets stars galaxies etc astrodynamics spacecraft navigation etc solid mechanics elasticity plasticity viscoelasticity exhibited deformable solids fracture mechanics acoustics sound density variation propagation solids fluids gases statics semi rigid bodies mechanical equilibrium fluid mechanics motion fluids soil mechanics mechanical behavior soils continuum mechanics mechanics continua solid fluid hydraulics mechanical


properties liquids fluid statics liquids equilibrium applied mechanics engineering mechanics biomechanics solids fluids etc biology biophysics physical processes living organisms relativistic einsteinian mechanics universal gravitation quantum following categorized part quantum mechanics schr dinger wave mechanics used describe movements wavefunction single particle matrix mechanics alternative formulation allows considering systems finite dimensional state space quantum statistical mechanics generalizes ordinary quantum mechanics consider systems unknown state often used derive thermodynamic properties particle physics motion structure


reactions particles nuclear physics motion structure reactions nuclei condensed matter physics quantum gases solids liquids etc historically classical mechanics around nearly quarter millennium quantum mechanics developed classical mechanics originated isaac newton 's laws motion philosophi naturalis principia mathematica developed seventeenth century quantum mechanics developed later nineteenth century precipitated planck 's postulate albert einstein 's explanation photoelectric effect fields commonly held constitute certain knowledge exists physical nature classical mechanics especially often viewed model


called exact sciences essential respect extensive use mathematics theories well decisive role played experiment generating testing quantum mechanics bigger scope encompasses classical mechanics sub discipline applies certain restricted circumstances according correspondence principle contradiction conflict two subjects simply pertains specific situations correspondence principle states behavior systems described quantum theories reproduces classical physics limit large quantum numbers ie quantum


mechanics applied large systems eg baseball result would almost classical mechanics applied quantum mechanics superseded classical mechanics foundation level indispensable explanation prediction processes molecular atomic sub atomic level however macroscopic processes classical mechanics able solve problems unmanageably difficult mainly due computational limits quantum mechanics hence remains useful well used modern descriptions behavior begin careful definition quantities displacement distance moved time velocity acceleration


mass force years ago however motion explained different point view example following ideas greek philosopher scientist aristotle scientists reasoned cannonball falls natural position earth sun moon stars travel circles around earth nature heavenly objects travel perfect circles often cited father modern science galileo brought together ideas great thinkers time began calculate motion terms distance travelled


starting position time took showed speed falling objects increases steadily time fall acceleration heavy objects light ones provided air friction air resistance discounted english mathematician physicist isaac newton improved analysis defining force mass relating acceleration objects traveling speeds close speed light newton 's laws superseded albert einstein 's theory relativity sentence illustrating computational complication einstein 's theory relativity atomic


subatomic particles newton 's laws superseded quantum theory everyday phenomena however newton 's three laws motion remain cornerstone dynamics study causes motion relativistic analogy distinction quantum classical mechanics albert einstein 's general special theories relativity expanded scope newton galileo 's formulation mechanics differences relativistic newtonian mechanics become significant even dominant velocity body approaches speed light instance newtonian mechanics kinetic energy free particle ''mv'' sup sup


whereas relativistic mechanics ''mc'' sup sup lorentz factor formula reduces newtonian expression low energy limit high energy processes quantum mechanics must adjusted account special relativity led development quantum field theory professional organizations applied mechanics division american society mechanical engineers fluid dynamics division american physical society society experimental mechanics http wwwimecheorg institution mechanical engineers united kingdom 's qualifying body mechanical engineers home mechanical engineers years http wwwiutamnet international union


applied mechanics https webarchiveorg web http wwwphysics onlinecom interactive learning resources teaching mechanics https webarchiveorg web http archimedesmpiwg berlinmpgde archimedes project articles containing video clips


'''mathematical logic''' study logic within mathematics major subareas include model theory proof theory set theory recursion theory research mathematical logic commonly addresses mathematical properties formal systems logic expressive deductive power however also include uses logic characterize correct mathematical reasoning establish foundations mathematics since inception mathematical logic contributed motivated study foundations mathematics study began late th century development axiomatic frameworks geometry arithmetic

Humanities
Philosophy
Logic
Mathematical logic


analysis early th century shaped david hilbert 's program prove consistency foundational theories results kurt g del gerhard gentzen others provided partial resolution program clarified issues involved proving consistency work set theory showed almost ordinary mathematics formalized terms sets although theorems cannot proven common axiom systems set theory contemporary work foundations mathematics often focuses establishing parts mathematics formalized particular formal systems reverse


mathematics rather trying find theories mathematics developed subfields scope ''handbook mathematical logic'' makes rough division contemporary mathematical logic four areas set theory model theory recursion theory proof theory constructive mathematics considered parts single area area distinct focus although many techniques results shared among multiple areas borderlines amongst fields lines separating mathematical logic fields mathematics always sharp g del 's incompleteness theorem marks milestone


recursion theory proof theory also led l b 's theorem modal logic method forcing employed set theory model theory recursion theory well study intuitionistic mathematics mathematical field category theory uses many formal axiomatic methods includes study categorical logic category theory ordinarily considered subfield mathematical logic applicability diverse fields mathematics mathematicians including saunders mac lane proposed category theory foundational system mathematics independent set theory foundations use


toposes resemble generalized models set theory may employ classical nonclassical logic history mathematical logic emerged mid th century subfield mathematics reflecting confluence two traditions formal philosophical logic mathematics mathematical logic also called 'logistic' 'symbolic logic' 'algebra logic' recently simply 'formal logic' set logical theories elaborated course last nineteenth century aid artificial notation rigorously deductive method emergence logic studied rhetoric ''calculationes'' syllogism


made philosophical mathematicians including leibniz lambert labors remained isolated little known th century middle nineteenth century george boole augustus de morgan presented systematic mathematical treatments logic work building work algebraists george peacock extended traditional aristotelian doctrine logic sufficient framework study foundations mathematics charles sanders peirce later built upon work boole develop logical system relations quantifiers published several papers


gottlob frege presented independent development logic quantifiers ''begriffsschrift'' published work generally considered marking turning point history logic frege 's work remained obscure however bertrand russell began promote near turn century two dimensional notation frege developed never widely adopted unused contemporary texts ernst schr der published ''vorlesungen ber die algebra der logik'' three volumes work summarized extended work boole de morgan peirce comprehensive reference


symbolic logic understood end th century foundational theories concerns mathematics built proper foundation led development axiomatic systems fundamental areas mathematics arithmetic analysis geometry logic term ''arithmetic'' refers theory natural numbers giuseppe peano published set axioms arithmetic came bear name peano axioms using variation logical system boole schr der adding quantifiers peano unaware frege 's work time around


time richard dedekind showed natural numbers uniquely characterized induction properties dedekind proposed different characterization lacked formal logical character peano 's axioms dedekind 's work however proved theorems inaccessible peano 's system including uniqueness set natural numbers isomorphism recursive definitions addition multiplication successor function mathematical induction mid th century flaws euclid 's axioms geometry became known addition independence parallel postulate established nikolai lobachevsky mathematicians discovered certain


theorems taken granted euclid fact provable axioms among theorem line contains least two points circles radius whose centers separated radius must intersect hilbert developed complete set axioms geometry building previous work pasch success axiomatizing geometry motivated hilbert seek complete axiomatizations areas mathematics natural numbers real line would prove major area research first half


th century th century saw great advances theory real analysis including theories convergence functions fourier series mathematicians karl weierstrass began construct functions stretched intuition nowhere differentiable continuous functions previous conceptions function rule computation smooth graph longer adequate weierstrass began advocate arithmetization analysis sought axiomatize analysis using properties natural numbers modern definition limit continuous functions already developed bolzano remained relatively unknown cauchy


defined continuity terms infinitesimals see cours d'analyse page dedekind proposed definition real numbers terms dedekind cuts rational numbers definition still employed contemporary texts georg cantor developed fundamental concepts infinite set theory early results developed theory cardinality proved reals natural numbers different cardinalities next twenty years cantor developed theory transfinite numbers series publications published new proof uncountability real numbers


introduced diagonal argument used method prove cantor 's theorem set cardinality powerset cantor believed every set could well ordered unable produce proof result leaving open problem th century early decades th century main areas study set theory formal logic discovery paradoxes informal set theory caused wonder whether mathematics inconsistent look proofs consistency hilbert


posed famous list problems next century first two resolve continuum hypothesis prove consistency elementary arithmetic respectively tenth produce method could decide whether multivariate polynomial equation integers solution subsequent work resolve problems shaped direction mathematical logic effort resolve hilbert 's ''entscheidungsproblem'' posed problem asked procedure would decide given formalized mathematical statement whether statement true false set theory paradoxes


burali forti first state paradox burali forti paradox shows collection ordinal numbers cannot form set soon thereafter bertrand russell discovered russell 's paradox jules richard discovered richard 's paradox zermelo provided first set axioms set theory axioms together additional axiom replacement proposed abraham fraenkel called zermelo fraenkel set theory zf zermelo 's axioms incorporated principle limitation size avoid russell 's paradox first volume ''principia mathematica'' russell alfred north whitehead published


seminal work developed theory functions cardinality completely formal framework type theory russell whitehead developed effort avoid paradoxes ''principia mathematica'' considered one influential works th century although framework type theory prove popular foundational theory mathematics fraenkel proved axiom choice cannot proved axioms zermelo 's set theory urelements later work paul cohen showed addition urelements needed axiom choice unprovable


zf cohen 's proof developed method forcing important tool establishing independence results set theory symbolic logic leopold l wenheim thoralf skolem obtained l wenheim skolem theorem says first order logic cannot control cardinalities infinite structures skolem realized theorem would apply first order formalizations set theory implies formalization countable model counterintuitive fact became known skolem 's paradox doctoral thesis kurt g del proved completeness theorem establishes correspondence syntax semantics first order logic g del


g del gave different consistency proof reduces consistency classical arithmetic intuitionistic arithmetic higher types first textbook symbolic logic layman written lewis carroll author alice wonderland beginnings branches alfred tarski developed basics model theory beginning group prominent mathematicians collaborated pseudonym nicolas bourbaki publish '' l ments de math matique'' series encyclopedic mathematics texts texts written austere axiomatic style emphasized rigorous presentation set theoretic foundations terminology coined texts


words ''bijection'' ''injection'' ' 'surjection'' set theoretic foundations texts employed widely adopted throughout mathematics study computability came known recursion theory computability theory early formalizations g del kleene relied recursive definitions functions definitions shown equivalent turing 's formalization involving turing machines became clear new concept ndash computable function ndash discovered definition robust enough admit numerous independent characterizations work incompleteness theorems g del lacked rigorous concept


core mathematical logic deals mathematical concepts expressed using formal logical systems systems though differ many details share common property considering expressions fixed formal language systems propositional logic first order logic widely studied today applicability foundations mathematics desirable proof theoretic properties stronger classical logics second order logic infinitary logic also studied along non classical logics intuitionistic logic first order logic '''first order logic''' particular formal system


logic syntax involves finite expressions well formed formulas semantics characterized limitation quantifiers fixed domain discourse early results formal logic established limitations first order logic l wenheim skolem theorem showed set sentences countable first order language infinite model least one model infinite cardinality shows impossible set first order axioms characterize natural numbers real numbers infinite structure isomorphism goal


early foundational studies produce axiomatic theories parts mathematics limitation particularly stark g del 's completeness theorem established equivalence semantic syntactic definitions logical consequence first order logic shows particular sentence true every model satisfies particular set axioms must finite deduction sentence axioms compactness theorem first appeared lemma g del 's proof completeness theorem took many years logicians grasped significance began apply routinely says


set sentences model every finite subset model words inconsistent set formulas must finite inconsistent subset completeness compactness theorems allow sophisticated analysis logical consequence first order logic development model theory key reason prominence first order logic mathematics g del 's incompleteness theorems establish additional limits first order axiomatizations '''first incompleteness theorem''' states consistent effectively given defined logical system capable interpreting arithmetic


exists statement true sense holds natural numbers provable within logical system indeed may fail non standard models arithmetic may consistent logical system example every logical system capable expressing peano axioms g del sentence holds natural numbers cannot proved logical system said effectively given possible decide given formula language system whether formula axiom one


express peano axioms called sufficiently strong applied first order logic first incompleteness theorem implies sufficiently strong consistent effective first order theory models elementarily equivalent stronger limitation one established l wenheim skolem theorem '' 'second incompleteness theorem''' states sufficiently strong consistent effective axiom system arithmetic prove consistency interpreted show hilbert 's program cannot reached classical logics many logics besides first order logic studied include infinitary logics allow formulas provide


infinite amount information higher order logics include portion set theory directly semantics well studied infinitary logic math l omega omega math logic quantifiers may nested finite depths first order logic formulas may finite countably infinite conjunctions disjunctions within thus example possible say object whole number using formula math l omega omega math math x lor x lor x lor cdots math higher order logics allow quantification elements


domain discourse subsets domain discourse sets subsets objects higher type semantics defined rather separate domain higher type quantifier range quantifiers instead range objects appropriate type logics studied development first order logic example frege 's logic similar set theoretic aspects although higher order logics expressive allowing complete axiomatizations structures natural numbers satisfy analogues completeness compactness theorems first order logic


thus less amenable proof theoretic analysis another type logics '' 's''' allow inductive definitions like one writes primitive recursive functions one formally define extension first order logic mdash notion encompasses logics section behave like first order logic certain fundamental ways encompass logics general eg encompass intuitionistic modal fuzzy logic lindstr 's theorem implies extension first order logic satisfying compactness theorem downward l wenheim skolem theorem first order logic nonclassical modal logic


modal logics include additional modal operators operator states particular formula true necessarily true although modal logic often used axiomatize mathematics used study properties first order provability set theoretic forcing intuitionistic logic developed heyting study brouwer 's program intuitionism brouwer avoided formalization intuitionistic logic specifically include law excluded middle states sentence either true negation true kleene 's work proof theory intuitionistic logic


showed constructive information recovered intuitionistic proofs example provably total function intuitionistic arithmetic computable true classical theories arithmetic peano arithmetic algebraic logic algebraic logic uses methods abstract algebra study semantics formal logics fundamental example use boolean algebras represent truth values classical propositional logic use heyting algebras represent truth values intuitionistic propositional logic stronger logics first order logic higher order logic studied using complicated algebraic structures


cylindric algebras set theory '''set theory''' study sets abstract collections objects many basic notions ordinal cardinal numbers developed informally cantor formal axiomatizations set theory developed first axiomatization due zermelo extended slightly become zermelo fraenkel set theory zf widely used foundational theory mathematics formalizations set theory proposed including von neumann bernays g del set theory nbg morse kelley set theory mk new foundations nf zf nbg


mk similar describing cumulative hierarchy sets new foundations takes different approach allows objects set sets cost restrictions set existence axioms system kripke platek set theory closely related generalized recursion theory two famous statements set theory axiom choice continuum hypothesis axiom choice first stated zermelo proved independent zf fraenkel come widely accepted mathematicians states given collection nonempty sets


single set ''c'' contains exactly one element set collection set ''c'' said choose one element set collection ability make choice considered obvious since set collection nonempty lack general concrete rule choice made renders axiom nonconstructive stefan banach alfred tarski showed axiom choice used decompose solid ball finite number pieces rearranged


scaling make two solid balls original size theorem known banach tarski paradox one many counterintuitive results axiom choice continuum hypothesis first proposed conjecture cantor listed david hilbert one problems g del showed continuum hypothesis cannot disproven axioms zermelo fraenkel set theory without axiom choice developing constructible universe set theory continuum hypothesis must hold paul cohen showed continuum hypothesis cannot


proven axioms zermelo fraenkel set theory independence result completely settle hilbert 's question however possible new axioms set theory could resolve hypothesis recent work along lines conducted w hugh woodin although importance yet clear contemporary research set theory includes study large cardinals determinacy large cardinals cardinal numbers particular properties strong existence cardinals cannot proved zfc existence smallest large cardinal typically studied inaccessible cardinal already implies


consistency zfc despite fact large cardinals extremely high cardinality existence many ramifications structure real line ''determinacy'' refers possible existence winning strategies certain two player games games said ' 'determined'' existence strategies implies structural properties real line polish spaces model theory '''model theory''' studies models various formal theories theory set formulas particular formal logic signature model structure


gives concrete interpretation theory model theory closely related universal algebra algebraic geometry although methods model theory focus logical considerations fields set models particular theory called elementary class classical model theory seeks determine properties models particular elementary class determine whether certain classes structures form elementary classes method quantifier elimination used show definable sets particular theories cannot complicated tarski established quantifier elimination real closed fields result


also shows theory field real numbers decidable also noted methods equally applicable algebraically closed fields arbitrary characteristic modern subfield developing concerned minimal structures morley 's categoricity theorem proved michael morley states first order theory countable language categorical uncountable cardinality ie models cardinality isomorphic categorical uncountable cardinalities trivial consequence continuum hypothesis complete theory less continuum many nonisomorphic countable models


countably many vaught 's conjecture named robert lawson vaught says true even independently continuum hypothesis many special cases conjecture established recursion theory '''recursion theory''' also called '''computability theory''' studies properties computable functions turing degrees divide uncomputable functions sets level uncomputability recursion theory also includes study generalized computability definability recursion theory grew work r zsa p ter alonzo church alan turing greatly


extended kleene post classical recursion theory focuses computability functions natural numbers natural numbers fundamental results establish robust canonical class computable functions numerous independent equivalent characterizations using turing machines lambda calculus systems advanced results concern structure turing degrees lattice recursively enumerable sets generalized recursion theory extends ideas recursion theory computations longer necessarily finite includes study computability higher types well areas hyperarithmetical


theory alpha recursion theory contemporary research recursion theory includes study applications algorithmic randomness computable model theory reverse mathematics well new results pure recursion theory algorithmically unsolvable problems important subfield recursion theory studies algorithmic unsolvability decision problem function problem '''algorithmically unsolvable''' possible computable algorithm returns correct answer legal inputs problem first results unsolvability obtained independently church turing showed entscheidungsproblem algorithmically unsolvable turing proved establishing


algorithmic unsolvability problem proved yuri matiyasevich proof theory constructive mathematics '''proof theory''' study formal proofs various logical deduction systems proofs represented formal mathematical objects facilitating analysis mathematical techniques several deduction systems commonly considered including hilbert style deduction systems systems natural deduction sequent calculus developed gentzen study '''constructive mathematics''' context mathematical logic includes study systems non classical logic intuitionistic logic well study predicative systems


early proponent predicativism hermann weyl showed possible develop large part real analysis using predicative methods proofs entirely finitary whereas truth structure common work constructive mathematics emphasize provability relationship provability classical nonconstructive systems provability intuitionistic constructive respectively systems particular interest results g del gentzen negative translation show possible embed ''translate'' classical logic intuitionistic logic allowing properties intuitionistic proofs


transferred back classical proofs recent developments proof theory include study proof mining ulrich kohlenbach study proof theoretic ordinals michael rathjen applications mathematical logic successfully applied mathematics foundations g frege b russell hilbert p bernays h scholz r carnap lesniewski skolem also physics r carnap dittrich b russell c e shannon n whitehead h reichenbach p fevrier biology j h woodger tarski psychology f b fitch c g hempel law morals k


menger u klug p oppenheim economics j neumann morgenstern practical questions e c berkeley e stamm even metaphysics j jan salamucha h scholz j bochenski applications history logic proven extremely fruitful j lukasiewicz h scholz b mates becker e moody j salamucha k duerr z jordan p boehner j bochenski stanislaw schayer ingalls applications also made theology f drewnowski j salamucha thomas connections computer science study computability theory computer science closely related


study computability mathematical logic difference emphasis however computer scientists often focus concrete programming languages feasible computability researchers mathematical logic often focus computability theoretical concept noncomputability theory semantics programming languages related model theory program verification particular model checking curry howard correspondence proofs programs relates proof theory especially intuitionistic logic formal calculi lambda calculus combinatory logic studied idealized programming languages computer science also contributes mathematics


developing techniques automatic checking even finding proofs automated theorem proving logic programming descriptive complexity theory relates logics computational complexity first significant result area fagin 's theorem established np precisely set languages expressible sentences existential second order logic foundations mathematics th century mathematicians became aware logical gaps inconsistencies field shown euclid 's axioms geometry taught centuries example axiomatic method incomplete use infinitesimals


definition function came question analysis pathological examples weierstrass' nowhere differentiable continuous function discovered cantor 's study arbitrary infinite sets also drew criticism leopold kronecker famously stated god made integers else work man endorsing return study finite concrete objects mathematics although kronecker 's argument carried forward constructivists th century mathematical community whole rejected david hilbert argued favor study infinite saying one shall expel us paradise cantor


created mathematicians began search axiom systems could used formalize large parts mathematics addition removing ambiguity previously naive terms function hoped axiomatization would allow consistency proofs th century main method proving consistency set axioms provide model thus example non euclidean geometry proved consistent defining ''point'' mean point fixed sphere ''line'' mean great circle sphere resulting structure


seriously affected g del 's incompleteness theorems show consistency formal theories arithmetic cannot established using methods formalizable theories gentzen showed possible produce proof consistency arithmetic finitary system augmented axioms transfinite induction techniques developed seminal proof theory second thread history foundations mathematics involves nonclassical logics constructive mathematics study constructive mathematics includes many different programs various definitions ''constructive'' accommodating end proofs


first stated order mathematical statement true mathematician person must able ''intuit'' statement believe truth understand reason truth consequence definition truth rejection law excluded middle statements according brouwer could claimed true negations also could claimed true brouwer 's philosophy influential cause bitter disputes among prominent mathematicians later kleene kreisel would study formalized versions


formal logics early th century ref ref name hilbertbernays plusnote foreword first edition grundlagen der mathematik bernays wrote following reminiscent famous note frege informed russell 's paradox blockquote style margin padding die ausf hrung dieses vorhabens hat eine wesentliche verz gerung dadurch erfahren da einem stadium dem die darstellung schon ihrem abschu nahe war durch das erscheinen der arbeiten von herbrand und von g del eine ver nderte situation im gebiet der beweistheorie entstand welche die ber cksichtigung neuer einsichten zur aufgabe machte dabei ist der umfang des buches angewachsen da eine


second volume included form gentzen 's consistency proof arithmetic ref references undergraduate texts shawn hedman ''a first course logic introduction model theory proof theory computability complexity'' oxford university press covers logics close relation computability theory complexity theory graduate texts kleene stephen cole '' https booksgooglecom booksid hzajpwaacaaj source gbs viewapi introduction metamathematics '' new york van nostrand ishi press reprint kleene stephen cole '' http worldcatorg oclc mathematical logic '' john wiley dover reprint research papers monographs texts surveys jd sneed ''the logical structure


mathematical physics'' reidel dordrecht revised edition br reprinted appendix classical papers texts collections reprinted english translation consistency irrational numbers two english translations ''essays theory numbers'' beman w w ed trans dover ''from kant hilbert source book foundations mathematics'' vols ewald william b ed oxford university press reprinted english translation notion 'definite' independence axiom choice frege gottlob ''begriffsschrift eine der arithmetischen nachgebildete


formelsprache des reinen denkens'' halle louis nebert translation ''concept script formal language pure thought modelled upon arithmetic'' bauer mengelberg frege gottlob ''die grundlagen der arithmetik eine logisch mathematische untersuchung ber den begriff der zahl'' breslau w koebner translation j l austin ''the foundations arithmetic logico mathematical enquiry concept number'' nd ed blackwell reprinted english translation gentzen 's ''collected works'' e szabo ed north holland amsterdam reprinted english translation g del 's ''collected works'' vol ii solomon feferman et al eds oxford university press


english edition ''the foundations geometry'' republished open court chicago lecture given international congress mathematicians september published english translation grounding elementary number theory mancosu pp nbsp reprinted english translation translated possibilities calculus relatives excerpt reprinted english translation principles arithmetic presented new method reprinted english translation principles mathematics problems sets reprinted english translation proof every set well ordered reprinted


set theory logic http wwwuclacuk philosophy lpsg philmathhtm philosophy mathematics


'''myth''' folklore genre consisting narratives play fundamental role society foundational tales origin myths main characters myths usually non humans gods demigods supernatural figures ref name mythfolk simpson jacqueline steve roud eds myths ''a dictionary english folklore'' oxford oxford university press ref ref name oflaherty ref ref name oxford literary however others also include humans animals combinations classification myth ref name ref stories everyday human beings although often leaders type usually contained legends opposed myths ref


name oxford literary myths sometimes distinguished legends myths deal gods usually historical basis set world remote past different present ref name oxford literary myths often endorsed secular religious authorities closely linked religion spirituality many societies group myths legends history together considering myths legends true accounts remote past ref name mythfolk particular creation myths take place primordial age world achieved later form myths


explain society 's customs institutions taboos established sanctified complex relationship recital myths enactment rituals term '''''mythology''''' may either refer study myths general body myths regarding particular subject study myth began ancient history rival classes greek myths euhemerus plato sallustius developed neoplatonists later revived renaissance mythographers today study myth continues wide variety academic fields including folklore studies philology psychology anthropology moreover


academic comparisons bodies myth known comparative mythology since term ''myth'' widely used imply story objectively true identification narrative myth highly controversial many adherents religions view religion 's stories true therefore object stories characterized myths seeing stories religions myth scholars label religious narratives ''myths'' practical reasons avoid depreciating one tradition cultures interpret differently relative one


another scholars avoid using term myth altogether instead utilize different terms like sacred history holy story simply history avoid placing pejorative overtones sacred narrative definitions arthurian mythology myth definitions ''myth'' vary extent among scholars though finnish folklorist lauri honko offers widely cited definition author title source another definition myth comes myth criticism theorist professor jos manuel losada according cultural myth criticism studies myth must understand explain global imaginary reality able better understand contemporary culture ref author title source scholars fields use


historical events th th centuries respectively became mythologised following centuries colloquial use word ''myth'' also used collectively held belief basis fact false story ref name https wwwlexicocom definition myth myth ''lexico'' oxford oxford university press retrieved may ref usage often pejorative arose labelling religious myths beliefs cultures incorrect spread cover non religious beliefs well however commonly used folklorists academics relevant fields anthropology term


''myth'' implication whether narrative may understood true otherwise among biblical scholars old new testament word myth technical meaning usually refers describe actions worldly terms world creation fall ref name oxford dic bib ref mythology mabinogi myths red book hergest written pre c incorporating pre roman myths celtic gods br ''gereint vab erbin arthur deuodes dala llys yg caerllion ar wysc'' br geraint son erbin arthur accustomed hold


court caerlleon upon usk present use ''mythology'' usually refers collected myths group people example greek mythology roman mythology celtic mythology hittite mythology describe body myths retold among cultures ref name grassie ''mythology'' also refer study myths mythologies mythography compilation description myths sometimes known ''mythography'' term also used scholarly anthology myths confusingly study myths generally key mythographers classical tradition include ovid bce ce whose tellings


myths profoundly influential fabius planciades fulgentius latin writer late th early th centuries whose ''mythologies'' gathered gave moralistic interpretations wide range myths anonymous medieval vatican mythographers developed anthologies classical myths remained influential end middle ages renaissance scholar natalis comes whose ten book ''mythologiae'' became standard source classical mythology later renaissance europe prominent mythographies include thirteenth century ''prose edda'' attributed icelander snorri sturluson main surviving survey norse mythology middle ages jeffrey g


myth' termed j r r tolkien amongst others refer conscious generation mythology notoriously also suggested separately nazi ideologist alfred rosenberg etymology '' odysseus overcome demodocus' song'' francesco hayez word ''myth'' comes ancient greek '' '' ''m thos'' meaning 'speech narrative fiction myth plot' anglicised form greek word began used english likewise adapted european languages early th century much narrower sense scholarly term traditional story especially one concerning early history


people explaining natural social phenomenon typically involving supernatural beings events ref name ref name oedmthos turn ancient greek ''mytholog a'' 'story' 'lore' 'legends' 'the telling stories' combines word ''m thos'' suffix '' '' '' logia'' 'study' order mean 'romance fiction story telling' ref name oedlogy logy ''comb form'' ''oxford english dictionary'' oxford oxford university press ref accordingly plato used ''mytholog a'' general term 'fiction' 'story telling' kind greek term ''mytholog a'' borrowed late latin occurring title latin author fulgentius' th century ''mythologi ''


denote call ''classical mythology'' ie greco roman etiological stories involving gods fulgentius' ''mythologi '' explicitly treated subject matter allegories requiring interpretation true events latin term adopted middle french ''mythologie'' whether french latin usage english adopted word ''mythology'' th century initially meaning 'the exposition myth myths' 'the interpretation fables' 'a book expositions' word first attested john lydgate 's ''troy book'' c ref name oedmlogy https wwwoedcom view entry mythology ''n'' ''oxford english dictionary'' rd ed oxford oxford university press accessed


aug ref br br wonderful expresse br schewed hym silf appearance br liche discriued fulgence br e book '''methologies''' lydgate th th century ''mythology'' used mean moral fable allegory parable collection traditional stories ref name oedmlogy edward dod london reprinted ref understood false came eventually applied similar bodies traditional stories among polytheistic cultures around world ref name oedmlogy thus word ''mythology'' entered english language word ''myth''


johnson 's ''dictionary'' example entry ''mythology'' ''myth'' w strahan london ref mythologist mythologize mythological mythologically indeed greek loanword ''mythos'' ''theology'' obviou ly evident mingling hi tory e men mortals came cribed gods would naturally occa ion sort generally find ''mythoi'' told pl ''mythoi'' latinate ''mythus'' year publisher w pickering pages ref pl ''mythi'' appeared english first example ''myth'' year publisher e typographia regia pages translated paraphrase year publisher printed year pages ref


two stories good illustrations origin ''myths'' means even natural sentiment traced cause circumstances fabulous history meanings ancient greece term ''m thos'' appears works homer poets homer 's era term several meanings 'conversation' 'narrative' 'speech' 'story' 'tale' 'word' similar related term ''logos'' ''mythos'' expresses whatever delivered form words contrasted greek ''ergon'' 'action' 'deed' 'work' however term ''mythos'' lacks


concerned gods heroes david wiles observes modern conceptions greek tragedy misleading commonly thought ancient audience members already familiar ''mythos'' behind play could predict outcome play however greek dramatists expected faithfully reproduce traditional myths adapting stage instead recreating myths producing new versions storytellers like euripides c bce relied suspense excite audiences one works merope attempts kill son 's murderer axe unaware


man question actually son according ancient description audience reactions work audience members genuinely unsure whether would commit filicide stopped time rose feet terror caused uproar david wiles points traditional ''mythos'' ancient greece primarily part oral tradition greeks era literate culture produced sacred texts definitive authoritative versions myths recorded texts preserved forever unchanging form


instead multiple variants myths circulation variants adapted songs dances poetry visual art performers myths could freely reshape source material new work adapting needs new audience response new situation children ancient greece familiar traditional myths early age according philosopher plato c bce mothers nursemaids narrated myths stories children charge david wiles describes repository mythological lore bruce lincoln called attention apparent


meaning terms ''mythos'' ''logos'' works hesiod ''theogony'' hesiod attributes muses ability proclaim truths narrate plausible falsehoods ie falsehoods seem like real things verb used narrating falsehoods text ''legein'' etymologically associated ''logos'' two variants manuscript tradition verb used proclaim truths one variant uses ''gerusasthai'' ''mythesasthai'' latter form verb ''mytheomai'' 'to speak' 'to tell' etymologically associated ''mythos'' ''works days''


hesiod describes dispute brother perses also announces readers intention tell true things brother verb uses telling truth ''mythesaimen'' another form ''mytheomai'' lincoln draws conclusion hesiod associated speech ''mythos'' lincoln calls telling truth associated speech ''logos'' telling lies hiding one 's true thoughts dissimulation conclusion strengthened use plural term ''logoi'' plural form ''logos'' elsewhere hesiod 's works three times term associated


term ' 'seductive'' three times term ''falsehoods'' genealogy gods hesiod lists ''logoi'' among children eris goddess personifying strife eris' children ominous figures personify various physical verbal forms conflict interpreting myths comparative mythology comparative mythology systematic comparison myths different cultures seeks discover underlying themes common myths multiple cultures cases comparative mythologists use similarities separate mythologies argue mythologies common source source may inspire myths provide


common protomythology diverged mythologies culture functionalism number commentators argued myths function form shape society social behaviour eliade argued one foremost functions myth establish models behavior myths may provide religious experience telling reenacting myths members traditional societies detach present returning mythical age thereby coming closer divine honko asserted cases society reenacts myth attempt reproduce conditions mythical age


example might reenact healing performed god beginning time order heal someone present similarly barthes argued modern culture explores religious experience since job science define human morality religious experience attempt connect perceived moral past contrast technological present pattanaik defines mythology subjective truth people communicated stories symbols rituals says facts everybody 's truth fiction nobody 's truth myths somebody 's truth euhemerism one theory claims myths


distorted accounts historical events according theory storytellers repeatedly elaborate upon historical accounts figures accounts gain status gods example myth wind god aeolus may evolved historical account king taught people use sails interpret winds herodotus fifth century bce prodicus made claims kind theory named ''euhemerism'' mythologist euhemerus c bce suggested greek gods developed legends human beings allegory theories propose myths began allegories natural phenomena


apollo represents sun poseidon represents water according another theory myths began allegories philosophical spiritual concepts athena represents wise judgment aphrodite desire ller supported allegorical theory myth believed myths began allegorical descriptions nature gradually came interpreted literally example poetic description sea raging eventually taken literally sea thought raging god personification thinkers claimed myths result personification objects forces according thinkers ancients


worshiped natural phenomena fire air gradually deifying example according theory ancients tended view things gods mere objects thus described natural events acts personal gods giving rise myths myth ritual theory according myth ritual theory myth tied ritual extreme form theory claims myths arose explain rituals claim first put forward smith argued people begin performing rituals reasons related myth forgetting original reason ritual account


inventing myth claiming ritual commemorates events described myth frazer argued humans started belief magical rituals later began lose faith magic invented myths gods reinterpreting rituals religious rituals intended appease gods history academic discipline historically important approaches study mythology included vico schelling schiller jung freud l vy bruhl l vi strauss frye soviet school myth ritual school ancient greece ''myths legends babylonia assyria'' critical interpretation myth began


presocratics euhemerus one important pre modern mythologists interpreted myths accounts actual historical events though distorted many retellings sallustius divided myths five categories theological physical concerning natural law animistic concerning soul material mixed concerns myths show interaction two previous categories particularly used initiations plato famously condemned poetic myth discussing education ''republic'' critique primarily grounds uneducated might take stories gods heroes literally nevertheless constantly referred


myths throughout writings platonism developed phases commonly called middle platonism neoplatonism writers plutarch porphyry proclus olympiodorus damascius wrote explicitly symbolic interpretation traditional orphic myths mythological themes consciously employed literature beginning homer resulting work may expressly refer mythological background without becoming part body myths cupid psyche medieval romance particular plays process turning myth literature ''euhemerism'' stated earlier refers rationalization myths putting themes formerly imbued mythological qualities


pragmatic contexts example would following cultural religious paradigm shift notably interpretation pagan mythology following christianization european renaissance panel bartolomeo di giovanni relates second half ''metamorphoses'' upper left jupiter emerges clouds order mercury rescue io interest polytheistic mythology revived renaissance early works mythography appearing sixteenth century among ''theologia mythologica'' nineteenth century v inen wise demigod one significant characters finnish mythological th century epic poetry ''the kalevala'' small ''v inen 's play'' robert wilhelm


ekman small first modern western scholarly theories myth appeared second half th century time word ''myth'' adopted scholarly term european languages ref name ref name oedmthos driven partly new interest europe 's ancient past vernacular culture associated romantic nationalism epitomised research jacob grimm movement drew european scholars' attention classical myths also material associated norse mythology finnish mythology forth western theories also partly driven europeans' efforts


comprehend control cultures stories religions encountering colonialism encounters included extremely old texts sanskrit ''rigveda'' sumerian ''epic gilgamesh'' current oral narratives mythologies indigenous peoples americas stories told traditional african religions ref name shippey shippey tom revolution reconsidered mythography mythology nineteenth century pp ''the shadow walkers jacob grimm mythology monstrous'' edited shippey tempe az arizona center medieval renaissance studies pp ref intellectual context nineteenth century scholars profoundly


shaped emerging ideas evolution ideas included recognition many eurasian languages therefore conceivably stories descended lost common ancestor indo european language could rationally reconstructed comparison descendant languages also included idea cultures might evolve ways comparable species ref name shippey general th century theories framed myth failed obsolete mode thought often interpreting myth primitive counterpart modern science within unilineal framework imagined human cultures travelling different speeds along linear path


cultural development nature mythology one dominant mythological theories latter th century ''nature mythology'' foremost exponents included max ller edward burnett tylor theory posited primitive man primarily concerned natural world tended interpret myths seemed distasteful european victorians tales sex incest cannibalism metaphors natural phenomena like agricultural fertility ref name mckinnell john ''meeting norse myth legend'' cambridge brewer pp ref unable conceive impersonal natural laws early humans tried explain natural phenomena


attributing souls inanimate objects thus giving rise animism according tylor human thought evolved stages starting mythological ideas gradually progressing scientific ideas ller also saw myth originating language even calling myth disease language speculated myths arose due lack abstract nouns neuter gender ancient languages anthropomorphic figures speech necessary languages eventually taken literally leading idea natural phenomena actuality conscious beings gods scholars even th century scholars accepted view however


lucien l vy bruhl claimed primitive mentality condition human mind stage historical development recent scholarship noting fundamental lack evidence nature mythology interpretations among people actually circulated myths likewise abandoned key ideas nature mythology ref name myth ritual james george frazer saw myths misinterpretation magical rituals based mistaken idea natural law idea central myth ritual school thought according frazer humans begin unfounded belief


imply modern humans must abandon myth twentieth century ''prometheus'' gustave moreau mythos hesiodus possibly aeschylus greek trilogy ''prometheus bound'' ''prometheus unbound'' ''prometheus pyrphoros'' prometheus bound tortured giving fire humanity earlier th century saw major work developing psychoanalytical approaches interpreting myth led sigmund freud drawing inspiration classical myth began developing concept oedipus complex ''the interpretation dreams'' jung likewise tried understand psychology behind world myths jung asserted humans share certain innate unconscious psychological forces


called ''archetypes'' believed similarities myths different cultures reveals existence universal archetypes mid th century saw influential development structuralist theory mythology led l vi strauss strauss argued myths reflect patterns mind interpreted patterns fixed mental structures specifically pairs opposites good evil compassionate callous rather unconscious feelings urges meanwhile bronislaw malinowski developed analyses myths focusing social functions real world associated idea myths origin stories might provide mythic charter legitimisation


relation history diverse social sciences studies share assumption history myth distinct sense history factual real accurate truth myth opposite barthes published series essays examining modern myths process creation book ''mythologies'' stood early work emerging post structuralist approach mythology recognised myths' existence modern world popular culture th century saw rapid secularisation western culture made western scholars willing


fragments partially lost myths th century structuralist work sought identify underlying patterns structures often diverse versions given myth tendency synthesise sources attempt reconstruct scholars supposed perfect underlying forms myths late th century however researchers influenced postmodernism tended instead argue account given myth cultural significance meaning argued rather representing degradation perfect form myths inherently plastic variable consequently


thing 'original version' 'original form' myth one prominent example movement k ramanujan 's essay three hundred ramayanas correspondingly scholars challenged precedence given texts medium mythology arguing media visual arts even landscape place naming could important br modern mythology belgian banknote depicting ceres neptune caduceus scholars field cultural studies research myth worked modern discourses mythological discourse reach greater audiences ever via


digital media various mythic elements appear television cinema video games although myth traditionally transmitted oral tradition small scale film industry enabled filmmakers transmit myths large audiences via film jungian psychology myths expression culture society goals fears ambitions dreams basis modern visual storytelling rooted mythological tradition many contemporary films rely ancient myths construct narratives walt disney company well known among cultural study scholars reinventing traditional childhood myths many films obvious


disney fairy tales plots many films based rough structure myths mythological archetypes cautionary tale regarding abuse technology battles gods creation stories often subject major film productions films often created guise cyberpunk action films fantasy dramas apocalyptic tales st century films ''clash titans'' ''immortals'' ''thor'' continue trend using traditional mythology frame modern plots authors use mythology basis books rick riordan whose percy jackson


olympians series situated modern day world greek deities manifest see also list mythologies list mythological objects list mythology books sources magic mythology mythopoeia artificially constructed mythology mainly purpose storytelling notes sources http searchebcom eb article myth ''encyclop dia britannica'' march year publisher university massachusetts press isbn translator last simpson translator first michael chapter introduction location amherst year publisher knopf canada isbn year publisher university california year publisher oxford university press isbn chapter euhemerism chapter url http wwwoxfordreferencecom views entryhtmlsubview main entry e year publisher clarendon press isbn year publisher penguin books limited isbn year publisher greenwood publishing group isbn year publisher continuum year publisher university virginia press isbn pages last dundes first alan chapter madness method plus plea projective inversion myth author link alan dundes


binary opposition myth propp levi strauss debate retrospect ''western folklore'' winter year publisher university california press isbn year publisher harvill press isbn translator first philip translator last mairet author link mircea eliade year publisher waveland press isbn author mask http wwwfupresscom archivio pdf pdf fabiani paolo philosophy imagination vico malebranche fup florence english edition pdf year publisher university chicago press isbn year publisher macmillan company limited pages year publisher johns hopkins university press isbn translator first thomas translator last marier year publisher university california press isbn year publisher university california press isbn pages year publisher taylor francis isbn northup lesley myth placed priorities religion study myth ''religious studies review'' year publisher oup oxford isbn pages simpson jacqueline steve roud eds myths


''a dictionary english folklore'' oxford oxford university press year publisher mit press isbn pages external links greek words phrases


mathematical logic '''model theory''' study relationship formal theories collection sentences formal language expressing statements mathematical structure models structures statements theory hold aspects investigated include number size models theory relationship different models interaction formal language particular model theorists also investigate sets defined model theory relationship definable sets

Humanities
Philosophy
Logic
Mathematical logic
Model theory


separate discipline model theory goes back alfred tarski first used term theory models publication since subject shaped decisively saharon shelah 's stability theory compared areas mathematical logic proof theory model theory often less concerned formal rigour closer spirit classical mathematics prompted comment '' proof theory sacred model theory profane '' applications model theory algebraic diophantine geometry reflect proximity classical mathematics


often involve integration algebraic model theoretic results techniques prominent scholarly organization field model theory association symbolic logic varied emphasis relative emphasis placed class models theory opposed class definable sets within model fluctuated history subject two directions summarised pithy characterisations respectively '''model theory''' universal algebra logic universal algebra stands mathematical structures logic logical theories '''model theory'''


algebraic geometry fields logical formulas definable sets equations varieties field nonetheless interplay classes models sets definable crucial development model theory throughout history instance stability originally introduced classify theories numbers models given cardinality stability theory proved crucial understanding geometry definable sets branches page focuses finitary first order model theory infinite structures finite model theory concentrates finite structures diverges significantly


g del 's completeness theorem upward downward l wenheim skolem theorems vaught 's two cardinal theorem scott 's isomorphism theorem omitting types theorem ryll nardzewski theorem examples early results model theory applied fields tarski 's elimination quantifiers real closed fields ax 's theorem pseudo finite fields robinson 's development non standard analysis important step evolution classical model theory occurred birth stability theory morley 's theorem uncountably categorical theories shelah 's classification program developed calculus independence rank based syntactical conditions satisfied theories last several


decades applied model theory repeatedly merged pure stability theory result synthesis called geometric model theory article taken include minimality example well classical geometric stability theory example proof geometric model theory hrushovski 's proof mordell lang conjecture function fields ambition geometric model theory provide ''geography mathematics'' embarking detailed study definable sets various mathematical structures aided substantial tools developed study pure model theory


fundamental notions first order model theory first order logic first order ''formula'' built atomic formulas ''r'' ''f'' ''x''''y'' ''z'' ''y'' ''x'' means boolean connectives math neg land lor rightarrow math prefixing quantifiers math forall v math math exists v math sentence formula occurrence variable scope corresponding quantifier examples formulas x mark fact x unbound variable defined follows math varphi forall u forall v exists w x times w u times v rightarrow exists w x times w u lor exists


w x times w v land x ne land x ne math math psi forall u forall v u times v x rightarrow u x lor v x land x ne land x ne math note equality symbol double meaning intuitively clear translate formulas mathematical meaning sub smr sub structure math mathcal n math natural numbers example element ''n'' ' 'satisfies'' formula ''n'' prime number formula similarly defines irreducibility tarski gave rigorous definition sometimes called tarski 's definition truth satisfaction relation math models math one easily proves math mathcal n models varphi n iff n math prime number


math mathcal n models psi n iff n math irreducible set ''t'' sentences called first order theory theory ' 'satisfiable'' ''model'' math mathcal models math ie structure appropriate signature satisfies sentences set ''t'' complete theory theory contains every sentence negation complete theory sentences satisfied structure also called ''theory structure'' g del 's completeness theorem confused incompleteness theorems says theory model


consistent ie contradiction proved theory therefore model theorists often use consistent synonym satisfiable basic model theoretic concepts signature language set non logical symbols symbol either function symbol relation symbol specified arity structure set math math together interpretations symbols signature relations functions math math confused interpretation one structure another common signature ordered rings math sigma times math math math


true math mathcal n math respect interpretation signature previously specified math mathcal n math substructure math mathcal math structure math mathcal b math subset domain closed functions signature regarded structure restricting functions relations subset generalises analogous concepts algebra instance subgroup substructure signature multiplication inverse substructure said ''elementary'' first order formula elements ''a'' sub sub


''a'' sub ''n'' sub math mathcal math math mathcal models varphi n math math mathcal b models varphi n math particular '' '' sentence math mathcal math elementary substructure math mathcal b math math mathcal models varphi math math mathcal b models varphi math thus elementary substructure model theory exactly superstructure model therefore field algebraic numbers math overline mathbb q math elementary substructure field complex numbers math mathbb c math rational field math mathbb q math express square root


first order sentence satisfied math mathbb c math math mathbb q math embedding structure math mathcal math another structure math mathcal b math map ''f'' ''a'' ''b'' domains written isomorphism math mathcal math substructure math mathcal b math written isomorphism elementary substructure called elementary embedding every embedding injective homomorphism converse holds signature contains relation symbols groups fields field vector space regarded


commutative group simply ignoring structure corresponding notion model theory ' 'reduct'' structure subset original signature opposite relation called ''expansion'' eg additive group rational numbers regarded structure signature expanded field signature times ordered group signature similarly ' signature extends another signature complete ' theory restricted intersecting set


sentences set formulas conversely complete theory regarded ' theory one extend one way complete ' theory terms reduct expansion sometimes applied relation well compactness l wenheim skolem theorem compactness theorem states set sentences satisfiable every finite subset satisfiable analogous statement ''consistent'' instead ' 'satisfiable'' trivial since every proof finite number antecedents used proof completeness theorem allows us transfer


elementarily embedded another structure cardinality straightforward generalisation uncountable signatures particular l wenheim skolem theorem implies theory countable signature infinite models countable model well arbitrarily large models certain sense made precise lindstr 's theorem first order logic expressive logic l wenheim skolem theorem compactness theorem hold definability definable sets model theory definable sets important objects study instance math mathbb n math formula math forall u forall v exists w x times w u times v rightarrow exists w x times w u lor exists w x times


w v land x ne land x ne math defines subset prime numbers formula math exists times x math defines subset even numbers similar way formulas ''n'' free variables define subsets math mathcal n math example field formula math x times x math defines curve math xy math math x math definitions mentioned ''parameter free'' defining formulas n't mention fixed domain elements however one also consider definitions ''with parameters model'' instance math mathbb r math formula math


x times x pi math uses parameter math pi math math mathbb r math define curve eliminating quantifiers general definable sets without quantifiers easy describe definable sets involving possibly nested quantifiers much complicated makes quantifier elimination crucial tool analysing definable sets theory ''t'' quantifier elimination every first order formula ''x'' sub sub ''x'' sub ''n'' sub signature equivalent modulo ''t'' first order formula ''x'' sub sub ''x'' sub ''n'' sub without quantifiers ie math forall x dots forall x n phi x dotsx n leftrightarrow psi x dotsx n math holds models ''t'' theory structure quantifier elimination


every set definable structure definable quantifier free formula parameters original definition example theory algebraically closed fields signature sub ring sub quantifier elimination means algebraically closed field every formula equivalent boolean combination equations polynomials theory quantifier elimination one add additional symbols signature early model theory spent much effort proving axiomatizability quantifier elimination results specific theories especially algebra often instead


quantifier elimination weaker property suffices theory ''t'' called model complete every substructure model ''t'' model ''t'' elementary substructure useful criterion testing whether substructure elementary substructure called tarski vaught test follows criterion theory ''t'' model complete every first order formula ''x'' sub sub ''x'' sub ''n'' sub signature equivalent modulo ''t'' existential first order formula ie formula following form math exists v dots exists v psi x dotsx nv dotsv math quantifier free theory


model complete may may ''model completion'' related model complete theory general extension original theory general notion ''model companion'' minimality every structure every finite subset math dots n math definable parameters simply use formula math x vee dots vee n math since negate formula every cofinite subset includes finitely many elements domain also always definable leads concept ''minimal structure'' structure


math mathcal math called minimal every subset math subseteq mathcal math definable parameters math mathcal math either finite cofinite corresponding concept level theories called ' 'strong minimality'' theory ''t'' called strongly minimal every model ''t'' minimal structure called ' 'strongly minimal'' theory structure strongly minimal equivalently structure strongly minimal every elementary extension minimal since theory algebraically closed fields quantifier elimination every definable subset algebraically closed field definable quantifier free formula one variable


quantifier free formulas one variable express boolean combinations polynomial equations one variable since nontrivial polynomial equation one variable finite number solutions theory algebraically closed fields strongly minimal hand field math mathbb r math real numbers minimal consider instance definable set math varphi x exists times x math defines subset non negative real numbers neither finite cofinite one fact use math varphi math define arbitrary intervals real number line turns


suffice represent every definable subset math mathbb r math generalisation minimality useful model theory ordered structures densely totally ordered structure math mathcal math signature including symbol order relation called minimal every subset math subseteq mathcal math definable parameters math mathcal math finite union points intervals definable interpretable structures particularly important definable sets also substructures e contain constants closed function application instance one study definable subgroups certain group