Minimal types in simple theories(5)

We prove that if M0 is a model of a simple theory, and p(x) is a complete type of Cantor-Bendixon rank 1 over M0, then p is stationary and regular. As a consequence we obtain another proof that any countable model M0 of a countable complete simple theory T

Regularityofp(x)isstandard.Butwegothroughtheproofforcom-pleteness.Wehavetoprove:

(*)IfMisamodelcontainingM0,brealizesaforkingextensionofpoverMandcrealizesa(orratherthe)nonforkingextensionofp(x)overMthencisindependentfromboverM.

Letdbethede ningschemaforp.Asbeforeletφ(x)isolatepamongnonalgebraictypesoverM0.AssumethatcforkswithMboverM0.Soforsomeψ(x,y,z)overM0,anda∈Mwehave|=ψ(c,b,a)∧¬dψ(b,a).AsbforkswithMoverM0wehavesomeχ(x,z )overM0anda ∈Msuchthat|=χ(b,a )∧¬dχ(a ).Sothefollowingholds:

y(φ(y)∧ψ(c,y,a)∧¬dψ(y,a)∧χ(y,a )∧¬dχ(a ))

Thereisnoharminassuminga=a .Astp(c/M0a)istheheiroftp(c/M0)=p,we nda0∈M0andb0suchthat

|=φ(b0)∧ψ(c,b0,a0)∧¬dψ(b0,a0)∧χ(b0,a0)∧¬dχ(a0)

Socforkswithb0overM0wherebyb0∈/M0,soasb0realizesφ,tp(b0/M0)=p(x).Butthenthefactthat|=χ(b0,a0)∧¬dχ(a0)givesacontradiction.Corollary2.3([4])IfTisacountablecompletesimpletheory,andM0acountablemodelofTthenM0hasin nitelymanycountableelementaryextensionsuptoisomorphismoverM0.

Proof.WemayassumeS(M0)tobecountablesocontainsaCB-rank1typep(x).Proposition2.1andCorollary2.2applytop.Leta1,..,anbeindependentrealizationsofp(overM0)andletMnbetheprimemodeloverM0∪{a1,..,an}.Ifb∈Mnrealizespthentp(b/M0,a1,..,an)isisolatedhencebythemoreoverclauseinCorollary2.2forkswitha1,..,anoverM0.Hence{a0,..,an}isamaximalindependent(overM0)setofrealizationsofpinMn.So(byregularity)thedimensionofpinMnisn.Thisconcludestheproof.Question1.SupposeM0isamodelofasimpletheory,andthateverytypeinS(M0)hasCB-rank.IseverycompletetypeoverM0stationary?Theresultsin[1]wereactuallyprovedunderaweakerassumptionthanstabilityofTh(M0).TheassumptionwasthatM0hasnoorder:thereisnoin nitesetoftuplesfromM0totallyorderedbysomeformula.

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