The Primal Framework II: Smoothness

Abstract Let ( K , ⩽, cpr) be a class of models with a notion of ‘strong’ submodel and of canonically prime model (cpr) over an increasing chain. We show under appropriate set-theoretic hypotheses that if K is not smooth (there are incompatible models over some chains), then K has many models in certain cardinalities. On the other hand, if K is smooth, we show that in reasonable cardinalities K has a unique homogeneous-universal model. In this situation we introduce the notion of type and prove the equivalence of saturated with homogeneous-universal.

[1]  Saharon Shelah,et al.  Finite diagrams stable in power , 1970 .

[2]  W. Hodges.  CLASSIFICATION THEORY AND THE NUMBER OF NON‐ISOMORPHIC MODELS , 1980 .

[3]  S. Shelah.  Classification of first order theories which have a structure theorem , 1985 .

[4]  Saharon Shelah,et al.  Categoricity of Theories in Lk omega, with k a Compact Ordinal , 1990, Annals of Pure and Applied Logic.

[5]  R. Vaught,et al.  Homogeneous Universal Models , 1962 .

[6]  Michael Makkai.  Categoricity of theories in L_ , with k a compact cardinal , 1990 .

[7]  Saharon Shelah,et al.  Classification theory - and the number of non-isomorphic models, Second Edition , 1990, Studies in logic and the foundations of mathematics.

[8]  Ami Litman,et al.  A Strengthening of Jensen's Principles , 1980, J. Symb. Log..

[9]  Ehud Hrushovski.  Finitely Based Theories , 1989, J. Symb. Log..

[10]  S. Shelah.  The spectrum problem I: ℵɛ-satuarted models, the main gap , 1982 .

[11]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[12]  Saharon Shelah,et al.  Remarks on squares , 1986 .

[13]  P. Dedecker.  Review: Roland Fraisse, Sur Quelques Classifications des Systemes de Relations , 1957 .

[14]  Victor Harnik,et al.  Review: S. Shelah, Classification Theory and the Number of Nonisomorphic Models , 1982, Journal of Symbolic Logic.

[15]  Bjarni Jónsson,et al.  Homogeneous Universal Relational Systems. , 1960 .

[16]  Saharon Shelah,et al.  The Primal Framework I , 1990, Ann. Pure Appl. Log..