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
Shelahโs 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
Waerdenโs 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 โต1-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๏ฌcient condition for a set ๐ท to have a nice dimension theory is that ๐ท be strongly
minimal or equivalently that algebraic closure on ๐ท 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 ๐ฟ๐1,๐.
Definition 1.1.1. A pregeometry is a set ๐บ together with a dependence relation
๐๐ : ๐ซ(๐บ) โ ๐ซ(๐บ)
satisfying the following axioms.
A1. ๐๐(๐) =
โช
{๐๐(๐โฒ): ๐โฒ โ๐๐๐ ๐}
A2. ๐ โ ๐๐(๐)
A3. ๐๐(๐๐(๐)) = ๐๐(๐)
A4. (Exchange) If ๐ โ ๐๐(๐๐) and ๐ โ โ ๐๐(๐), then ๐ โ ๐๐(๐๐).
If points are closed the structure is called a geometry.
Definition 1.1.2. A geometry is homogeneous if for any closed ๐ โ ๐บ and
๐,๐ โ ๐บ โ ๐ there is a permutation of ๐บ which preserves the closure relation (i.e.
an automorphism of the geometry), fixes ๐ pointwise, and takes ๐ to ๐.
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http://dx.doi.org/10.1090/ulect/050/01
