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1、TerminologyProposalsforaHyperoperationEnvironmentHelloeverybody!IampleasedtobeinvolvedinthisForumfromitsbeginning,sinceIamconvincedthatthesubjectitcoverswillsurelybecomeofgreatimportanceinthefuturedevelopmentsofmathematics.Nevertheless,IshouldliketoproposethattheForumadoptsthesi
2、mplestandclearestapproach,forinvolvingthemaximumnumberofpeople,whilekeepingthenecessarystrictrigorousmathematicalformalizationtoasecondphase,ortospecializedsub-forums.Clearandpracticalterminologicalchoicesarethereforeindispensable.Herearemyproposals,asaresultofalongcooperationwi
3、thKonstantinRubtsov(KAR).Theseconventionswereinfactusedinareportthatwejointlypreparedin2003(publishedontheWebon17-09-2004)andmentionedinthebibliographyoftheStephenWolframsbookANewKindofScience.See,forinstance:http://forum.wolframscience.com/showthread.php?s=a62c97b0b7fa64d17ffee
4、94adf2eb8b7&threadid=579http://forum.wolframscience.com/showthread.php?s=a62c97b0b7fa64d17ffee94adf2eb8b7&threadid=956Infact,inanextremelysimple,informalandintuitiveway,wecandefineaparticularternaryoperation,i.e.involvingthreenumbers,asfollows(theoperationbecomesbinaryforanycons
5、tants):Ha,b
6、scwith:a:thefirstoperand,orthebaseb:thesecondoperand,ortheexponentorlevelordegrees:the(ternaryoperation)ranksuchthat:Ha,a
7、sHa,2
8、s1and:Ha,Ha,a
9、s
10、sHa,3
11、s1Byusinganinfixedoperator,wecould,moreeasilyandclearly,write:asbcasaas12asasaas13and:asasas....asa....as1nntimesWese
12、ethatanyoperationofranks,appliedonitsbaseanditeratedntimes,givesanoperationofranks+1,withexponent(level,degree)equalton.ItcanbeshownthatthisternaryoperationcanbeputinbijectionwiththeelementsoftheAckermannfunction,ifweputa1babab.Inthiscase,weobtainaninfinitehierarchyofoperations(
13、calledtheGrzegorczykHiierarchy),threeofwhicharethewellknownbinaryoperationofclassicalalgebra.Infact,weget:s=1a1babadditions=2a2bababababmultiplicationbs=3a3ba^baexponentiationIndeed,wecanverifythat:aaa...aa...anadditionmultiplicationnnaaa...aa...a^namultiplicationexponentiationn
14、na^a^a^...a^a...anaexponentiationtetrationn1The
