CHAPTER 1 Combinatorial Geometries and Infinitary Logics In this chapter we introduce two of the key concepts that are used throughout the text. In the first section we define the notion of a combinatorial geometry and describe its connection with categoricity in the first order case. The second section establishes the basic notations for infinitary logics. 1.1. Combinatorial Geometries Shelahโ€™s notion of forking provides a notion of independence which greatly generalizes the notion of Van der Waerden [VdW49] by allowing one to study structures with a family of dimensions. (See Chapter II of [Bal88].) But Van der Waerdenโ€™s notion is the natural context for studying a single dimension in another guise such a relation is called a combinatorial geometry. Most of this monograph is devoted to structures which are determined by a single dimension in the simplest case, which we study first, the entire universe is the domain of a single combinatorial geometry. One strategy for the analysis of โ„ต1-categorical first order theories in a countable vocabulary proceeds in two steps: a) find a definable set which admits a nice dimension theory b) show this set determines the model up to isomorphism. The su๏ฌƒcient condition for a set ๐ท to have a nice dimension theory is that ๐ท be strongly minimal or equivalently that algebraic closure on ๐ท forms a pregeometry in the sense described below. It is natural to attempt to generalize this approach to study categoricity in non-elementary contexts. In this chapter we review the first order case to set the stage. For more detail on this chapter consult e.g. [Bue91]. In the next few chapters, we study the infinitary analog to the first order concept of strong minimality โ€“ quasiminimality. In Part 4 we show that the strategy of reducing categoricity to a quasiminimal excellent subset can be carried out for sentences of ๐ฟ๐œ” 1 ,๐œ” . Definition 1.1.1. A pregeometry is a set ๐บ together with a dependence relation ๐‘๐‘™ : ๐’ซ(๐บ) โ†’ ๐’ซ(๐บ) satisfying the following axioms. A1. ๐‘๐‘™(๐‘‹) = โˆช {๐‘๐‘™(๐‘‹โ€ฒ): ๐‘‹โ€ฒ โŠ†๐‘“๐‘–๐‘› ๐‘‹} A2. ๐‘‹ โŠ† ๐‘๐‘™(๐‘‹) A3. ๐‘๐‘™(๐‘๐‘™(๐‘‹)) = ๐‘๐‘™(๐‘‹) A4. (Exchange) If ๐‘Ž โˆˆ ๐‘๐‘™(๐‘‹๐‘) and ๐‘Ž โˆ• โˆˆ ๐‘๐‘™(๐‘‹), then ๐‘ โˆˆ ๐‘๐‘™(๐‘‹๐‘Ž). If points are closed the structure is called a geometry. Definition 1.1.2. A geometry is homogeneous if for any closed ๐‘‹ โŠ† ๐บ and ๐‘Ž,๐‘ โˆˆ ๐บ โˆ’ ๐‘‹ there is a permutation of ๐บ which preserves the closure relation (i.e. an automorphism of the geometry), fixes ๐‘‹ pointwise, and takes ๐‘Ž to ๐‘. 3 http://dx.doi.org/10.1090/ulect/050/01
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