A Guide to Classical and Modern Model Theory

Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the Wroclaw Logic colloquium 2007. The goal was to survey recent work in model theory of o-minimal structures, centered around the solution to a beautiful conjecture of Pillay on definable groups in o-minimal structures. The conjecture (which is now a theorem in most interesting cases) suggested a connection between arbitrary definable groups in o-minimal structures and compact real Lie groups.

We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures: Theorem. Let \mathcal{M} be an o-minimal expansion of a real closed field, \langle G;+\rangle a 2-dimensional group definable in \mathcal{M} , and \mathcal{D}=\langle G;+,\ldots\rangle a strongly minimal structure, all of whose atomic relations are definable in \mathcal{M} . If \mathcal{D} is not locally modular, then an algebraically closed field K is interpretable in \mathcal{D} , and the group G , with all its induced \mathcal{D} -structure, is definably isomorphic in \mathcal{D} to an algebraic K -group with all its induced K -structure.

An argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

In this paper we will explain some basic notions related to quantifier elimination in the first order theories. We will give a general algorithm for quantifier elimination for any theory. The examples of theories which admit QE are theory of dense linear order (DLO), theory of algebraically closed fields (ACF) and theory of real closed fields (RCF). At the end, we will show the applications of quantifier elimination in ACF and RCF. The interesting applications can be seen in geometry, biology and control theory.

Some model-theoretic results in the algebraic theory of quadratic forms

It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal).It is simultaneously proved that if M is 0minimal, then every definable set of n-tuples of M has finitely many "definably connected components."0. Introduction.In this paper we study the structure of definable sets (of tuples) in an arbitrary O-minimal structure M (in which the underlying order is dense).Recall from [PSI, PS2] that the structure M is said to be O-minimal if M = (M, <, Ri)iei, where < is a total ordering on M and every definable (with parameters) subset of M is a finite union of points in M and intervals (a, b) where aE M or a = -co and b E M or b = +co.M is said to be strongly O-minimal if every N which is elementarily equivalent to M is O-minimal.We will always assume that the underlying order of M is a dense order with no first or least element.In this paper we also introduce the notion of a definable set X C Mn being definably connected, and we prove THEOREM 0.1.Let M be O-minimal.Then any definable X C Mn is a disjoint union of finitely many definably connected definable sets.THEOREM 0.2.If M is O-minimal, then M is strongly O-minimal.THEOREM 0.3.(a) Let M be O-minimal and let (p(xi,...,xn,yi,...,ym) be any formula of L (the language for M).Then there is K < uj such that for any b E Mm, the set <j>(x,b)M (= {5 E Mn:M t= 4>(a,b)}) has at most K definably connected components.(b) If M is a O-minimal expansion o/(R, <), then in (a) we can replace definably connected by connected.Theorems 0.1 and 0.2 are proved simultaneously by a rather complicated induction argument (outlined in §3 and undertaken in § §4 and 5).Theorem 0.3 follows from Theorems 0.1 and 0.2 by a compactness argument.Let us remark that if M is the field of real numbers, or more generally any real closed field, then by Tarski's quantifier elimination [T], M is (strongly) O-minimal and moreover the definable sets (of n-tuples) in M are precisely the semialgebraic

This paper presents verified quantifier elimination procedures for dense linear orders (DLO), for real and for integer linear arithmetic. The DLO procedures are new. All procedures are defined and verified in the theorem prover Isabelle/HOL, are executable and can be applied to HOL formulae themselves (by reflection).

Suppose T is superstable and P is a complete type over some finite set with U(p) = a + 1 for some a.We show how to associate with p an incidence geometry which measures the complexity of the family of extensions of p of rank a.When p is stationary we give a characterization of the possible incidence geometries. As an application we proveTheorem.Suppose M is superstable and has only one l-type p e S(0).Further suppose p is stationary with U(p) = a + 1 for some a.Then one of the following holds: (i) There is an equivalence relation E c M2 with infinitely many infinite classes definable over 0.(ii) M is the algebraic closure of a set of Morley rank 1.In particular, M is Unstable of finite rank.

Let ℜ be an expansion of a dense linear order (R, &lt;) without endpoints having theintermediate value property, that is, for alla, b∈R, every continuous (parametrically) definable functionf: [a, b] →Rtakes on all values inRbetweenf(a) andf(b). Every expansion of the real line (ℝ, &lt;), as well as every o-minimal expansion of (R, &lt;), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, &lt;) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10).Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, &lt;, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: IfGis a proper nontrivial subgroup of (R, *) definable in ℜ, then eitherGis dense and codense inR, orGcontains an elementusuch that (R, &lt;, *,e, u, G) is elementarily equivalent to (ℚ, &lt;, +, 0, 1, ℤ), whereedenotes the identity element of (R, *) (Theorem 2.3).Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, &lt;, +, 0, 1, ℤ).

Abstract.We prove weak elimination of imaginary elements for Boolean orderings with finitely many atoms. As a consequence we obtain equivalence of the two notions of o-minimality for Boolean ordered structures, introduced by C. Toffalori. We investigate atoms in Boolean algebras induced by algebraically closed subsets of Boolean ordered structures. We prove uniqueness of prime models in strongly o-minimal theories of Boolean ordered structures.