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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
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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)
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