Minimal types in simple theories(4)

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

(iv)r(x)∈S(M0c)isanonforkingextensionofp(x)butisnot nitelysat-is ableinM0.

BytheIndependenceTheoremoveramodel(forsimpletheories),wecan ndbrealizingbothq(x)andr(x)(suchthatmoreover{a,c,b}isM0-independent).

By(iii)letψ(x,y)beaformulaoverM0suchthat

(v)|=ψ(b,a)∧¬ψ(c0,a).

By(iv)letχ(x,z)beaformulaoverM0suchthat

(vi)|=χ(b,c)andχ(x,c)isnotrealizedinM0.

Letusnow xaformulaφ(x)overM0whichisolatespamongnonalgebraictypesoverM0(whichexistsaspisassumedtohaveCB-rank1).Soby(v)and(vi)weclearlyhave

|=¬ψ(c0,a)∧( x)(φ(x)∧χ(x,c)∧ψ(x,a))

By(ii)thereisa inM0suchthat

( )|=¬ψ(c0,a )∧( x)(φ(x)∧χ(x,c)∧ψ(x,a ))

Soletb ly

|=φ(b )∧χ(b ,c)∧ψ(b ,a )

By(vi)andourassumptiononφweseethattp(b /M0)=p(x).Soψ(x,a )∈p(x).Ontheotherhandby( )and(i),¬ψ(x,a )∈p(x).Thisisacontradictionandprovestheproposition.

Corollary2.2LetM0andp(x)∈S(M0)beasinProposition2.1.Thenp(x)isde nableandregular.Moreoverthenonforkingextensionsofp(x)arepreciselythecoheirsofp(x).

Proof.Themoreoverclauseisclearfromstationarityofp:foranysetA M0phasanonforkingextensionswhichisacoheir,souseuniqueness.

De nabilityisalsoclear:p(x)hasauniqueheiroveranyset(itsnonfork-ingextension)sobyBethde nability,pisde nable(alternativelysee[3]).MoreoverclearlythenonforkingextensionofpoveranyA M0isgivenbyapplyingthede ningschemaofptoA.

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