Ranks and pregeometries in finite diagrams

Annals of Pure and Applied Logic 106 (1-3):49-83 (2000)
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Abstract

The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the 0-stable case which share many of the formal properties of forking. This is achieved by considering a rank for this framework which is bounded when the diagram D is 0-stable. We can also obtain pregeometries with respect to this dependence relation. The dependence relation is the natural one induced by the rank, and the pregeometries exist on the set of realizations of types of minimal rank. Finally, these concepts are used to generalize many of the classical results for models of a totally transcendental first-order theory. In fact, strong analogies arise: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are different, as we do not have compactness. This is illustrated with positive results as well as negative results . We also give a proof of a Two Cardinal Theorem for this context

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References found in this work

One hundred and two problems in mathematical logic.Harvey Friedman - 1975 - Journal of Symbolic Logic 40 (2):113-129.
Categoricity for abstract classes with amalgamation.Saharon Shelah - 1999 - Annals of Pure and Applied Logic 98 (1-3):261-294.
On strongly minimal sets.J. T. Baldwin & A. H. Lachlan - 1971 - Journal of Symbolic Logic 36 (1):79-96.
The primal framework II: smoothness.J. T. Baldwin & S. Shelah - 1991 - Annals of Pure and Applied Logic 55 (1):1-34.
The primal framework I.J. T. Baldwin & S. Shelah - 1990 - Annals of Pure and Applied Logic 46 (3):235-264.

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