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Dependent \(T\) and existence of limit models. (English) Zbl 1335.03033

It is well-known that saturated models of a complete first-order theory are unique up to isomorphism given their cardinality. The author generalized this to abstract elementary classes, considering the so-called \((\lambda,\kappa)\) limit models for pairs of cardinals \(\kappa<\lambda\). In this paper, he studies the question when such models exist in a complete first-order theory.
Recall that a universal model \(M\models T\) of cardinality \(\lambda\) is a \((\lambda,\kappa)\) superlimit model if for all ordinals \(\delta<\lambda^+\) of cofinality \(\kappa\) and for all \(\prec\)-continuous increasing sequences \((M_i:i\leq\delta)\) with \(M_i\cong M\) for all \(i<\delta\), one has \(M_\delta\cong M\). More generally, if there is a function \(F\) from the collection of models of \(T\) of cardinality \(\lambda\) to itself satisfying \(N\prec F(N)\) for all \(N\), such that
(a)
for all ordinals \(\delta<\lambda^+\) of cofinality \(\kappa\) and all \(\prec\)-continuous increasing sequences \((M_i:i\leq\delta)\) with \(F(M_{i+1})\prec M_{i+2}\) for all \(i<\delta\) one has \(M\cong\bigcup_{i<\delta}M_i\), then \(M\) is a \((\lambda,\kappa)\) strong limit;
(b)
for all \(\prec\)-continuous increasing sequences \((M_i:i<\lambda^+)\) with \(F(M_{i+1})\prec M_{i+2}\) for all \(i<\lambda^+\) there is a closed unbounded subset \(C\subseteq\lambda^+\) such that \(M\cong M_\delta\) for all \(\delta\in C\) of cofinality \(\kappa\), then \(M\) is a \((\lambda,\kappa)\) normal limit.
If \(F\) takes \(\prec\)-increasing continuous sequences as argument, and we replace the condition \(M_{i+1}\prec F(M_{i+1})\prec M_{i+2}\) by \(M_{i+1}\prec F(M_j:j\leq i+1)\prec M_{i+2}\), we obtain the notion of \((\lambda,\kappa)\) medium limit from (a) and \((\lambda,\kappa)\) weak limit from (b). If we replace \(M_{i+1}\prec F(M_{i+1})\prec M_{i+2}\) by \(M_{i+1}\prec N\prec M_{i+2}\) for some \(N\equiv_{M_{i+1}}F(M_{i+1})\) (and similarly for sequences), we obtain the notions of \((\lambda,\kappa)\) invariantly strong/normal/medium/weak limit. Clearly, superlimit implies strong limit, which implies both medium and normal limit, either of which implies weak limit. There are also variants limit\(^+\), limit\(^-\), and for replacing the set of ordinals of cofinality \(\kappa\) by some arbitrary stationary subset of \(\lambda^+\), but this would go beyond the limits of this review.
The author shows that
(1)
The theory of dense linear orders has \((\lambda,\kappa)\) invariantly medium limits, but no \((\lambda,\kappa)\) superlimit.
(2)
Strongly independent theories have no \((\lambda,\kappa)\) invariantly medium limit; independent theories have no \((\lambda,\kappa)\) medium limit.
He conjectures that dependent theories have \((\lambda,\kappa)\) invariantly medium limits.

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C55 Set-theoretic model theory
06A05 Total orders
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