Berkeley Model Theory Seminar

A vector space $V$ with the closure operation defined by taking a subset $A \subseteq V$ to the linear span of $A$ is a quintessential example of a pregeometry.  To any vector space $V$ we may define the dimension of $V$ to be cardinality of a basis, a maximal linearly independent set.   It follows from Zorn’s Lemma that bases exist and some simple manipulations permit one to see that all bases have the same cardinality so that $\dim V$ is a well-defined cardinal.

We may transpose the definition of a basis to any pregeometry and then the proofs of the existence of bases and that they all have the same cadinality lift from linear algebra to general pregeometries.

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For the sake of building a dimension theory for definable sets in o-minimal structures, we will  follow a more model theoretic approach, due to Pillay, to dimensions than what appears in van den Dries’ book.  That is, we will show that o-minimal structures are geometric which means that algebraic closure endows the universe of such a structure with a pregeometry, or what is sometimes called a matroid.  In the following post we will explain how to define dimensions in pregeometries.

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In our proof of the Cell Decomposition Theorem, we took as known the Uniform Finiteness Theorem.  In this post, we fill that gap.

Uniform Finiteness Theorem:   Let $Y \subseteq R^{n+1}$ be a definable set for which all fibers $Y_a := \{ y \in R : (a,y) \in R \}$ are finite as $a$ ranges through $R^n$.  Then there is a number $N = N(Y)$ so that all fibers have size at most $N$.

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The Cell Decomposition Theorem might be called the “Fundamental Theorem of O-minimality”.   With this theorem we show that from the hypothesis that the definable sets in one variable are particularly simply, i.e. those which admit a quantifier-free definition in the reduct to the language of ordered sets with parameters, it follows that the higher dimensionsal sets are also tame.

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A basic principle in tame geometry is that there are no pathological definable functions in o-minimal structures.   One precise sense in which this principle is true is given by the Monotonicity Theorem.

Monotonicity Theorem :  Let $f: R \to R$ be a definable function in some o-minimal structure $\mathfrak{R}$.  Then $f$ is piecewise continuous and is piecewise constant or strictly monotone.  That is, we can find $-\infty = a_0 < a_1 < \ldots < a_m = \infty$ so that for $0 \leq i < m$ the function $f \upharpoonright (a_i,a_{i+1})$ is continuous and constant, strictly increasing, or strictly decreasing.

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For the most part, we will be following Lou van den Dries’ book Tame Topology and O-minimal Structures for this seminar.  We will supplement with material from reseach papers.  If there are specific theorems about o-minimality you would like to see this term, let me know.  Some possible topics include

• Speissegger’s theorem on Pfaffian closures of o-minimal structures
• The Peterzil-Starchenko o-minimal trichotomy
• The Pila-Wilkie counting theorem
• Proofs of o-minimality of specific structures (e.g. $\mathbb{R}_{an,\exp}$ or the expansion by Gevrey functions)
• The Peterzil-Starchenko o-minimal Chow theorem

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