## Independence, dimension and continuity in non-forking frames.(English)Zbl 1315.03048

Summary: The notion $$J$$ is independent in $$(M,M_0,N)$$ was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal $$\lambda$$ and has a non-forking relation, satisfying the good $$\lambda$$-frame axioms and some additional hypotheses. Shelah uses independence to define dimension.
Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved.
As a corollary, we weaken the stability hypothesis and two additional hypotheses, that appear in Shelah’s theorem.

### MSC:

 03C45 Classification theory, stability, and related concepts in model theory 03C48 Abstract elementary classes and related topics
Full Text:

### References:

 [1] Fundamentals of stability theory (1988) · Zbl 0685.03024 [2] DOI: 10.1016/0168-0072(84)90004-6 · Zbl 0588.03014 [3] Classification theory for abstract elementary classes (2009) · Zbl 1225.03036 [4] Classification theory for abstract elementary classes (2009) · Zbl 1225.03036 [5] Classification theory and the number of nonisomorphic models 92 (1978) [6] Linear orderings 98 (1982) [7] DOI: 10.1090/conm/302/05080 [8] Model theory for infinitary logic (1971) [9] DOI: 10.1016/S0168-0072(97)00019-5 · Zbl 0897.03036 [10] DOI: 10.1016/0168-0072(89)90059-6 · Zbl 0697.03024 [11] Annals of Pure and Applied Logic 215 pp 245– (1984) [12] Shelah’s categoricity conjecture from a successor for tame abstract elementary classes 71 pp 553– (2006) · Zbl 1100.03023 [13] DOI: 10.1142/S0219061306000487 · Zbl 1107.03029 [14] Logic and its applications pp 203– (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.