Logical analysis of special relativity theory

LogialanalysisofspeialrelativitytheoryHajnalAndreka,JuditX.Madarasz,andIstvanNemeti1IntrodutionInthispaperwetrytogiveasmallsampleillustratingtheapproahofAndrekaetal.[2℄toalogialanalysisofrelativitytheoryondutedpurelyinrstorderlogi(formethodologialreasons).Hereweonentrateonspeialrelativity,butin[2℄stepsaremadeinthediretionofgeneralizingtheapproahtowardsgeneralrelativity.In[2℄webuildupvariantsofrelativitytheoryas\ompetingaxiomsystemsformalizedinrstorderlogi.Thereasonforhavingseveralversionsforthetheory,i.e.severalaxiomsystems,isthatthiswayweanstudytheonsequenesofthevariousaxioms,enablingustondoutwhihaxiomisresponsibleforsomeinterestingor\exotipreditionofrelativitytheory.Amongothers,thisenablesustorenetheoneptualanalysisinFriedman[6℄andRindler[11℄,oromparetheReihenbah-Grunbaumapproahtorelativity(f.L.E.Szabo[13℄or[6℄)withthestandardone.Asexplainedin[2℄,thepresentapproahis(insomesense)moreambitious(asarelativitytheory)thane.g.aformalizationof,say,Minkowskiangeometryinrstorderlogiwouldbe,invariousrespets:(i)OnerespetisthatifweidentiedMinkowskiangeometrywithspeialrelativity,thenthiswouldyieldanuninterpreted(inthephysialsense)versionofrelativity,whiletherstor-dertheorywhihwedevelopin[2℄ontains\itsowninterpretation,too.(ii)Itisnotleartoushowtheoneptualanalysis1suggestede.g.in[6℄(ortheReihenbah-Grunbaumissues)ouldbesqueezedintoMinkowskiangeometry.(iii)Ourformalizedrelativitytheoryisundeidable,whiletherstorderver-sionofMinkowskiangeometryin[7℄isdeidable,pointinginthediretionthatperhapsinourtheoryoneantalkaboutthingswhihdonotappearinthepureMinkowskiangeometry.SomeonemayarguethatMinkowskiangeome-tryistheheartofspeialrelativitytheory,butitisonlytheheart;andwewouldliketoformalizethefulltheoryandnotonlyitsheart.(iv)Theobser-vational/theoretialdualityoutlinedin[6℄motivatesustokeepourvoabularyandaxiomsontheobservationalside(whileMinkowskiangeometryremainsmoreonthe\theoretialside).Afterhavingformalizedrelativityinrstorderlogi,oneanusethewelldevelopedmahineryofrstorderlogiforstudyingpropertiesofthetheorySupportedbyHungarianNationalFoundationforSientiResearhgrantsT30314,T23234.1Whihaxiomisresponsibleforwhat,whihaxiomisintuitivelymorenaturalthantheother,et.1(e.g.thenumberofnon-elementarilyequivalentmodels,oritsrelationshipswithGodel’sinompletenesstheorems,independeneissues,et).TheideasinJohan’s[5℄(ombinedwiththeonesintheversionofSain[12℄updatedduringtheA’dam-Budapestooperation)explainwhywehavetoinsistonkeepingouraxiomatirelativitytheorywithin(possiblymodalandmany-sorted)rstorderlogi.ForamoreomprehensiveintrodutionandformoreonnetionswithJohan’s(orTarski’s,Suppes’,Goldblatt’set)workwereferto[2℄.Inpassingwenotethatinseveralrespetswearealsomotivatedbyideassimilartothosesummarized,inanilluminatingway,initems1,2ofthePrefaeofthebookMatolsi[8℄.2TheframelanguageWeintroduetherstorderlanguage,whihwewilluseforformalizing(speial)relativity.Wewanttotalkaboutmotionofbodies.2Whatismotion?Itishangingloationintime.Thereforewewilltalkaboutbodies,time,spae,andaboutaloation-funtionwhihtellsuswhihbodyiswhereatagiventime.Wewanttotalkaboutrelativitytheories;thereforetheseloationfuntionswilldependonobservers;dierentobserversmayseethesamemotiondierently.(Theloationfuntiondeterminedbyanobservermwillbealledtheworld-viewfuntionwmofobserverm.)Wewilltreatobserversasspeialbodieswhosemotionwill(ofourse)berepresentedexatlythesamewayasthatoftherestofthebodies.Theseobserversareoftenalled,intheliterature,refereneframes.3Itwillbeonvenientforustobeexibleaboutthedimensionofspae:wewillnotonlytreat3-dimensionalspae,but1or2,orhigher-dimensionalspaesaswell.Wewilltreattimeasaspeialdimensionofspae-time.nwilldenotethedimensionofourspae-time.Thus,usuallyn=4(3spae-dimensionsand1time-dimension),butwewillonsideralson=2;3orn4.Ourbodieswillbeidealized,pointlike.4Thevoabularyofourlanguageisthefollowing:unaryrelationsB(bodies)Obs(observers)Ph(photons)2Inthispaperweonentrateonlyonkinematis;thesamekindofinvestigationsanbearriedoutonerningmass,fores,energyet.Butifatheoremanbeprovedwithoutreferringtotheseextranotions,weonsiderthatavirtue.3Thediereneisonlyamatterofterminologyandwedonotnditimportantfromthepointofviewofthepresentwork.4Fromthepointofviewofthequestionsstudiedherethisdoesnotrestritgenerality.Ifsomereaderwouldprefer\fatobserversto\thinobservers,weanalwaysidentifyanobservermwiththe\refereneframeinduedbym,andthenthiswillyielda\fatnotionofobserver.2Q(quantitiesusedforgivingloationand\measuringtime);ann+2-aryrelation,theloation-orworld-viewrelationW(world-viewrelation,W(m;b;t;s1;:::;sn1)intendstomeanthata-ordingtoobserver(orreferene-frame)m,thebodybispresentattimetandloation(s1;:::;sn1));fordealingwithquantities,wewillhavetwobinaryfuntions,andabinaryrelation:+;;.Inourtheories,wewillalwaysassumethefollowing:observersandphotonsarebodiesW(m;b;t;s1;:::;sn1)impliesthatmisanobserver,bisabody,andt;s1;:::;sn1arequantities(Q;+;;)isaEulidean5linearlyorderedeld.Wefoundthatthesimplestwayoftreatingtheseassumptionsistousea2-sortedrst-orderlanguage,wher