TY - JOUR
T1 - Classification theory for non-elementary classes I
T2 - The number of uncountable models of ψ ∈L ω_1, ω. Part A
AU - Shelah, Saharon
PY - 1983/9
Y1 - 1983/9
N2 - Assuming that 2Nn < 2Nn+1 for n < ω, we prove that every ψ ∈L ω_1, ω has many non-isomorphic models of power N n for some n>0 or has models in all cardinalities. We can conclude that every such Ψ has at least 2 N 1 non-isomorphic uncountable models. As for the more vague problem of classification, restricting ourselves to the atomic models of some countable T (we can reduce general cases to this) we find a cutting line named "excellent". Excellent classes are well understood and are parallel to totally transcendental theories, have models in all cardinals, have the amalgamation property, and satisfy the Los conjecture. For non-excellent classes we have a non-structure theorem, e.g., if they have an uncountable model then they have many non-isomorphic ones in some N n (provided {ie212-7}).
AB - Assuming that 2Nn < 2Nn+1 for n < ω, we prove that every ψ ∈L ω_1, ω has many non-isomorphic models of power N n for some n>0 or has models in all cardinalities. We can conclude that every such Ψ has at least 2 N 1 non-isomorphic uncountable models. As for the more vague problem of classification, restricting ourselves to the atomic models of some countable T (we can reduce general cases to this) we find a cutting line named "excellent". Excellent classes are well understood and are parallel to totally transcendental theories, have models in all cardinals, have the amalgamation property, and satisfy the Los conjecture. For non-excellent classes we have a non-structure theorem, e.g., if they have an uncountable model then they have many non-isomorphic ones in some N n (provided {ie212-7}).
UR - http://www.scopus.com/inward/record.url?scp=51249186829&partnerID=8YFLogxK
U2 - 10.1007/BF02761954
DO - 10.1007/BF02761954
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AN - SCOPUS:51249186829
SN - 0021-2172
VL - 46
SP - 212
EP - 240
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
IS - 3
ER -