Andrej bauer - PDF document

Externalandinternalview Comparisonofconceptsasviewedbyus(externally)andbymathematiciansinsideEff(internally): Symbol External Internal N naturalnumbers naturalnumbers R computablereals allreals f:N!N computablemap anymap e:NA computableenumerationofA anyenumerationofA ftrue;falseg truthvalues decidabletruthvalues truthvaluesofEff truthvalues 8x computablyforallx forallx 9x thereexistscomputablex thereexistsx P_:P decisionprocedureforP PornotP Outline Introduction ConstructiveMathematics ComputabilitywithoutAxioms AxiomofEnumerability MarkovPrinciple TheTopologicalView RecursionTheorem InseparableSets Conclusion Intuitionisticlogic I Weuseintuitionisticlogic,morepreciselytheinternallanguageofatopos. I WhatisthestatusofLawofExcludedMiddle(LEM)?8p2 :(p_:p)“Foreverypropositionp,pornotp.”Inintuitionisticmathematicsitcanonlybeusedinspecialcases,whenpisdecidable. I Atthispointwedonotknowwhetherallpropositionsaredecidable,butlateroneofouraxiomswillfalsifyLEM. I ThestatusoftheAxiomofChoicewillbediscussedlater. Basicsetsandconstructions I Basicsets:;;1=fg;N=f0;1;2;:::g I Setoperations:AB;A+B;BA=A!B;fx2Ap(x)g;PA I WesaythatAis I non-emptyif:8x2A:?, I inhabitedif9x2A:�. AxiomofChoice I AxiomofChoice: Everyf:ABhasachoicefunctiong:A!Bsuchthatg(x)2f(x)forallx2A. ThiswedonotacceptbecauseitimpliesLEM. I WeacceptNumberChoice: Everyf:NBhasachoicefunctiong:N!B. I WealsoacceptDependentChoice: Givenx2Aandh:AA,thereexistsg:N!Asuchthatg(0)=xandg(n+1)2h(g(n))foralln2N. Thisisaformofsimplerecursionformulti-valuedfunctions. Decidableandclassicalsets I AsubsetSAisequivalentlygivenbyitscharacteristicmapS:A! ;S(x)=(x2S): I AsubsetSAisdecidableifS:A!2,equivalently8x2A:(x2S_x62S): I AsubsetSAisclassicalifS:A! ::,equivalently8x2A:(:(x62S)=)x2S): Outline Introduction ConstructiveMathematics ComputabilitywithoutAxioms AxiomofEnumerability MarkovPrinciple TheTopologicalView RecursionTheorem InseparableSets Conclusion Non-enumerabilityofCantorandBairespace Arethereanysetswhicharenotenumerable? Yes,forexamplePN,andalso: Corollary 2NandNNarenotenumerable. Proof. 2andNdonothavethexed-pointproperty. Wehaveprovedourrstsynthetictheorem: Theorem(externaltranslationofabovecorollary) Thesetofrecursivesetsandthesetoftotalrecursivefunctionscannotbecomputablyenumerated. ProjectionTheorem Theorem(Projection) AsubsetofNisenumerableiffitistheprojectionofadecidablesubsetofNN. Proof. IfAisenumeratedbye:N!1+AthenAistheprojectionofthegraphofe,fhm;ni2NNm=e(n)g: IfAistheprojectionofBNN,denee:NN!1+Abyehm;ni=ifhm;ni2Bthenmelse?: SemidecidablesubsetsofN Theorem TheenumerablesubsetsofNarethesemidecidablesubsetsofN. Proof. AnenumerableANistheprojectionofadecidableBNN.Thenn2Aiff9m2N:hn;mi2B.Conversely,ifA2N,byNumberChoicethereisd:NN!2suchthatn2Aiff9m2N:d(m;n). TheenumerablesubsetsofN:E=N:Note:atthispointwedonotknowwhetherE=PN. Partialfunctions I Givenasingle-valuedRB,thecorrespondingf:A!PBalwaysfactorsthrougheB=fS2PB8x;y2B:(x2S^y2S=)x=y)g: I Thuspartialmapsf:A*Barejustordinarymapsf:A!eB. I Writef(x)#whenfisdenedatx,i.e.,9y2B:y2f(x). Domainsof-partialfunctions Thesupport(a.k.a.domain)off:A*Bisfx2Af(x)#g. Proposition Asubsetissemidecidableiffitisthesupportofa-partialfunction. Proof. AsemidecidablesubsetS2AisthedomainofitscharacteristicmapS:A!=1?.Conversely,iff:A!B?is-partialthenitsdomainisthesetfx2Af(x)#g,whichisobviouslysemidecidable. Theorem(Externaltranslation) Asetissemidecidableiffitisthedomain(support)ofapartialcomputablemap. AxiomofEnumerability Axiom(Enumerability) Thereareenumerablymanyenumerablesetsofnumbers. LetW:NEbeanenumeration. Proposition andEhavethexed-pointproperty. Proof. ByLawvere,W:NE=N=NN=EN. ImmuneandSimpleSets I Asetisimmuneifitisneithernitenorinnite. I Asetissimpleifitisopenanditscomplementisimmune. Theorem ThereexistsanimmunesubsetofN. Proof. FollowingPost,considerP=fhm;ni2NNn�2m^n2Wmg,andletf:N!N?beaselectionforP.WeclaimthatS=im(f)=fn2N9m2N:f(m)=ngissimpleandNnSimmune.Becausef(m)�2mthesetNnScannotbenite.ForanyinniteenumerablesetUNnSwithU=Wm,wehavef(m)#,f(m)2Wm=U,andf(m)2S,henceUisnotcontainedinNnS. EndofPartI Walkaroundandrestyourbrainfor10minutes. MarkovPrinciple I Ifabinarysequencea22Nisnotconstantly0,doesitcontaina1? I Forp2,doesp6=?implyp=�? I Is ::? Axiom(MarkovPrinciple) Abinarysequencewhichisnotconstantly0containsa1. Phoa'sprinciple Whatdoes!looklike? Theorem(Phoa'sPrinciple) Foreveryf:!andx2,f(x)=(f(?)_x)^f(�): TheproofusesEnumerationaxiomandMarkovPrinciple.Theprinciplesaysthat!isaretractofwith I section:f7!hf(?);f(�)i I retraction:(u;v)7!x::(u_x)^v Aconsequenceismonotonicityoff:!:ifxythenf(x)=(f(?)_x)^f(�)(f(?)_y)^f(�)=f(y): TheTopologicalView I Thetopologicalview:semidecidablesubsets=opensubsets: I istheSierpinskispace:thespaceontwopoints?,�with�fgopenandf?gclosed. I ThetopologyofAisA. I “Allfunctionsarecontinuous.” Givenanyf:A!BandU2B,thesetf�1(U)isopenbecauseitisclassiedbyUf:A!. Thegenericconvergentsequence I Theone-pointcompacticationofNisN+=fa:N!28n2N:anan+1g: I Anaturalnumbernisrepresentedby0;0;:::;0| {z }n;1;1;::: I Innity1correspondsto0;0;0;::: I isaquotientofN+byq:N+,q(a)=(a1)=(9n2N:an=1): Thetopologyofan!-cpo A!-cpoisaposet(P;)inwhichincreasingchainshavesuprema. Theorem AnopensubsetU:P!is I upwardclosed:x2U^xy=)y2U I inaccessiblebychains:givenachaina:N!P,ifWkak2Uthenak2Uforsomek2N. Proof. (a)givenx2Uandxy,denef:N+!Pbyf(u)=_k2Nifkuthenxelsey:Thenx=f(1)2Uhenceforsomeu1wehavey=f(u)2U.(b)Similarly,considerf(u)=Wk2Namin(k;u). Outline Introduction ConstructiveMathematics ComputabilitywithoutAxioms AxiomofEnumerability MarkovPrinciple TheTopologicalView RecursionTheorem InseparableSets Conclusion Enumerablefocalsets I Enumerablefocalsets,knownasErsovcompletesets,havegoodproperties. I AatdomainA?isfocal.ItisenumerableifAisdecidableandenumerable. I IfAisenumerableandfocalthensoisAN:N'//// NN?eN?//// AN?NA//// AN I SomeenumerablefocalsetsareN;2N?;NN?: ClassicalRecursionTheorem Corollary(ClassicalRecursionTheorem) Foreveryf:N!Nthereisn2Nsuchthat'f(n)='n. Proof. InRecursionTheorem,taketheenumerablefocalsetA=NN?andthemulti-valuedfunctionF(g)=fh2NN?9n2N:g='n^h='f(n)g:Thereisgsuchthatg2F(g).Thusthereexistsn2Nsuchthat'n=g=h='f(n). Plotkin'sDomain2N? I Inapartiallyorderedset(P;)wesaythatxandyareincomparableifx6yandy6x. I Musttherealwaysbeamaximalelementaboveanelementofaposet? I Thesetof-partialbinaryfunctionsN!2?isapartiallyordered:fg()8n2N:f(n)g(n):ThisisPlotkin'suniversaldomain. Conclusion I Thetheme:weshouldlookforelegantpresentationsofstructureswestudy.Theycanleadtonewintuitions(anddestroyoldones). I Theseslides,andmore,atmath.andrej.com. References I Berger,U.,Totalsetsandobjectsindomaintheory,AnnalsofPureandAppliedLogic60(1993),pp.91-117 I Bridges,D.,Richman,F.,VarietiesofConstructiveMathematics,LectureNotesSer.,97,LondonMath.Soc.,1987. I Fenstad,J.,Onaxiomatizingrecursiontheory,in:J.E.Fenstadtetal.,editor,GeneralizedRecursionTheory,NorthHolland,1974pp.385-404. I Friedman,H.,Axiomaticrecursivefunctiontheory,in:Gandyetal.,editor,LogicColloquium'69(1971),pp.113-137. I Hyland,J.,Theeffectivetopos,in:A.TroelstraandD.V.Dalen,editors,TheL.E.J.BrouwerCentenarySymposium(1982),pp.165-216. I Hyland,J.,Firststepsinsyntheticdomaintheory,in:CategoryTheory,number1488inLectureNotesinMathematics,1991. I Moschovakis,Y.,Axiomsforcomputationtheories-rstdraft,in:Gandyetal.,editor,LogicColloquium'69(1971),pp.199-255. I Richman,F.,Church'sthesiswithouttears,TheJournalofSymbolicLogic48(1983),pp.797-803. I Rosolini,G.,ContinuityandEffectivenessinTopoi,Ph.D.thesis,UniversityofOxford(1986). I Spreen,D.,Oneffectivetopologicalspaces,TheJournalofSymbolicLogic63(1998),pp.185-221.