Some Themes Around First Order Theories Without the Independence Property
The aim of these notes (as well as the course of lectures they are based on) is to describe some current work around theories with NIP (not the independence property). This is a broad class of first order theories, including natural examples such as algeb
- PDF / 322,664 Bytes
- 21 Pages / 439.36 x 666.15 pts Page_size
- 33 Downloads / 203 Views
1 Model Theory In this first section I discuss model theory and notation, in a form appropriate to the subsequent sections. In the subsequent sections I will draw a lot on the papers [7], [8] and [9] as well as on papers of Chernikov and Simon [3, 4]. I fix a “language” or “vocabulary” L consisting of relation symbols Ri (each of a given arity 1), function symbols fj (each of a given arity 1) and constant symbols ck . We always include a distinguished binary relation “D”. If you want, take the language as being countable. From these symbols, together with the logical connectives ^; _; !; :; 9; 8, parentheses, and a collection of variables, we build up inductively the class of first order L-formulas. For a finite tuple xN D .x1 ; ::; xn / of variables, .x/ N denotes an L-formula whose free variables are among x1 ; ::; xn . By an L-sentence I mean an L-formula with no free variables. An L-structure M consists of an underlying set (often notationally identified with M ), and interpretations Ri .M /, fj .M /, ck .M / of the relation symbols, function symbols, constant symbols of L, where the equality symbol is interpreted as equality. So for example if Ri has arity ni , then Ri .M / is a subset of M ni . One could also work with a many-sorted language, where the symbols come together with sorts (so e.g. a ni -ary relation symbol Ri comes equipped with an ni -tuple of sorts). An L-structure will then be a collection of underlying sets indexed by the sorts and the symbols are interpreted accordingly. I will normally assume L to be 1-sorted, but there is a certain construction (the eq construction) taking us naturally into the many-sorted framework.
A. Pillay () University of Leeds, Leeds, UK University of Notre Dame, Notre Dame, IN, USA e-mail: [email protected] L. van den Dries et al., Model Theory in Algebra, Analysis and Arithmetic, Lecture Notes in Mathematics 2111, DOI 10.1007/978-3-642-54936-6__2, © Springer-Verlag Berlin Heidelberg 2014
13
14
A. Pillay
For an L-structure M , an L-formula .x/ N and an n-tuple aN of elements of (the underlying set of) M , we obtain, by an inductive definition, the notion “M ˆ .a/”: N .x/ N is true of aN in M , or aN satisfies .x/ N in M . When is an L-sentence, we say is true in M , or M is a model of , for M ˆ . If † is a collection of L-sentences, we write M ˆ † for “M ˆ for all 2 †” and again say “M is a model of †”. I will not work with any notion of “formal proof”, just “logical implication”. Definition 1.1. Let † be a set of L-sentences and an L-sentence. We write † ˆ , and say † (logically) implies , if any model of † is a model of . The most basic fact about first order logic is the Compactness Theorem. Theorem 1.2. Let †; be as above. Then † ˆ if and only if †0 ˆ for some finite subset †0 of †. Equivalently a collection † of sentences has a model iff every finite subset of † has a model. I sometimes say “† is consistent” for “† has a model”. Definition 1.3. (i) An L-theory is a consistent set of L-sentences, closed under logical implicati
Data Loading...
