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Title
Abstract
Introduction
Millar Asked the Following Question
References

Pure computable model theory

Profile image of Valentina HarizanovValentina Harizanov

1998

Abstract

Let ψ s = ∧Ψ s . If ψ s = ψ s (c 0 , . . . , c n s ), then for every e ∈ {0, . . . , n s }, we set ψ s e = def ∃y e+1 . . . ∃y n s ψ s (c 0 , . . . , c e , y e+1 , . . . , y n s ). For every e ≥ 0, at almost every stage s of the construction, we have a type Ω s e ∈ {Γ n : n ∈ ω} which is a candidate for a principal type realized by ([c 0 ], . . . , [c e ]). We will allow Ω s e to be undefined for finitely many s. Because of the consistency property, if Ω s e is defined then ψ s e (x/c) ∈ Ω s e . The construction will satisfy the following requirements for every e ≥ 0. P 1 e : σ e ∈ Ψ or ¬σ e ∈ Ψ; P 2 e : If σ e ∈ Ψ and σ e = ∃xθ(x), then θ(c) ∈ Ψ for some c ∈ C; Q e : ([c 0 ], . . . , [c e ]) realizes a principal type of T . The priority ranking of the requirements in the decreasing order is: P 1 0 , P 2 0 , Q 0 , . . . , P 1 e , P 2 e , Q e , . . . We attempt to satisfy the requirements in the order of their priority. We say that at stage s > 0: P 1 e requires attention if σ e / ∈ Ψ s-1 and ¬σ e / ∈ Ψ s-1 ; P 2 e requires attention if σ e ∈ Ψ s-1 and σ e = ∃xθ(x) for some θ such that θ(c) / ∈ Ψ s-1 for every c ∈ C; Q e requires attention if Ω s-1 e is undefined. Once satisfied at some stage, requirements P 1 e and P 2 e are never injured again. However, we say that Q e is injured at stage s > 0 if Ω s-1 e

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