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1、THEWUNKNOABLEG.J.ChaitinIBMhResearcDivisionT.J.atsonWResearchCenterHawthorne,NewYorkJanuary3,1999PrefaceHavingpublishedfourbooksonthissubject,whya fth?!Becausethere'ssomethingnew:IcompareandcontrastGodel's,uring'sTandmyworkinaverysimpleandstraight-for
2、wardmannerusing.LISPUptonowIneverwantedtoexamineGodel'sanduring'sTworktooclosely
3、Iwantedtodevelopmyownviewpoint.Butthereisnolongeranydanger.SoIsetouttoexplainthemathematicalessenceofthreeverydi erentwaystoexhibitlimitstomathematicalreasoning:thewayGod
4、elanduringTdiditinthe1930s,andmywaythatI'vebeenworkingonsincethe1960s.Inanutshell,Godeldiscoveredincompleteness,uringTdiscovered,uncomputabilityandIdiscoveredrandomness
5、that'stheamazingfactthatsomemathematicalstatementsaretruefornoreason,they'retrueby
6、accident.Therecanbenotheoryofeverything,"atleastnotinmathematics.Maybeinphysics!Ididn'twanttowriteajournalistic"book.Iwantedtoexplainthefundamentalmathematicalideas.understandablyAndIthinkthatI'vefoundawaytodoit,andthatIunderstandmypredecessors'work
7、betterthanIdidbefore.Theessenceofthisbookiswords,explainingmathematicalideas,butreaderswhofeelsoinclinedcanfollowmeallthewaytoLISPprogramsthatprettymuchshowGodel's,uring'sTandmyproofsworkingonthecomputer.Andifyouwanttoplaywiththissoftware,youcandownlo
8、aditfrommywebsite.Thisbookisalsoaprequel"tomySpringerbookTheLimitsofMathematics.It'saneasierintroductiontomyideas,andusesthesameversionofLISPthatIuseinmySpringervolume.Hopefullyit'llbea34steppingstoneforthoseforwhomTheLimitsofMathematicsistoointimida
9、ting.ThisbookbeganasalectureonAhundredyearsofcontroversyregardingthefoundationsofmathematics"inwhichIsummarizedtheworkofCantor,Russell,Hilbert,Godel,uring,Tandmyown,andconcludedthatmathematicsisquasi-empirical.Igaveseveraldi erentversionsofthislectur
10、einSpanishduringtwodelightfulvisitstoBuenosAiresin1998sponsoredbytheFacultyofExactSciencesoftheUni-versityofBuenosAires.ThesevisitswerearrangedbyProf.GuillermoMartnezoftheMathematicsDepartmentandbyProf.eronicaVBecheroftheComputerScienceDepar
