## 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

### Keywords:

limit model; dense linear order; strong dependence; dependence
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### References:

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