TY - JOUR
T1 - Strongly dependent theories
AU - Shelah, Saharon
N1 - Publisher Copyright:
© 2014, Hebrew University of Jerusalem.
PY - 2014/10
Y1 - 2014/10
N2 - We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ⊆ ℭ and |I|≥ℶ|T|+(µ), then some J ⊆ I of cardinality µ+ is an indiscernible sequence over A.
AB - We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:715], [Sh:783] and related works. Those are properties (= classes) somewhat parallel to superstability among stable theory, though are different from it even for stable theories. We show equivalence of some of their definitions, investigate relevant ranks and give some examples, e.g., the first order theory of the p-adics is strongly dependent. The most notable result is: if |A| + |T| ≤ µ, I ⊆ ℭ and |I|≥ℶ|T|+(µ), then some J ⊆ I of cardinality µ+ is an indiscernible sequence over A.
UR - http://www.scopus.com/inward/record.url?scp=84939878520&partnerID=8YFLogxK
U2 - 10.1007/s11856-014-1111-2
DO - 10.1007/s11856-014-1111-2
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AN - SCOPUS:84939878520
SN - 0021-2172
VL - 204
SP - 1
EP - 83
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 1
ER -