Minimal types in simple theories

时间:2025-07-14

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

Minimaltypesinsimpletheories

AnandPillay

UniversityofLeeds

September4,2006

Abstract

WeprovethatifM0isamodelofasimpletheory,andp(x)isacompletetypeofCantor-Bendixonrank1overM0,thenpisstation-aryandregular.AsaconsequenceweobtainanotherproofthatanycountablemodelM0ofacountablecompletesimpletheoryThasin- nitelymanycountableelementaryextensionsuptoM0-isomorphism.Thelatterextendsearlierresultsoftheauthorinthestablecase,andisaspecialcaseofarecentresultofTanovic[4].

1Introduction

Thispaper,whichextendsearlywork[1]oftheauthor,iscloselyrelatedtoandmotivatedbycurrentworkofPredragTanovicontheauthor’soldcon-jecturethatanycountablemodelM0(inacountablelanguage)hasin nitelymanycountablemodelsuptoisomorphismoverM0.In[4]Tanovicprovestheconjecturefortheorieswithoutthestrictorderproperty,andinprivatee-maildiscussionshehasdescribedaroutetothefullconjecture.

Theexpression“minimaltype”inthetitlereferstoatypeofCB-rank1overamodelM0,ratherthantoatypeofSU-rank1.Forp(x)∈S(M0)tobeofCantor-Bendixonrank1meansthatp(x)lythereissomeformulaφ(x)overM0suchthatp(x)isaxiomatizedby{φ(x)}∪{x=a:a∈ SupportedbyaMarieCurieChair

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