A Course in Model Theory

Abstract

This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions. The authors introduce the classic results, as well as more recent developments in this vibrant area of mathematical logic. Concrete mathematical examples are included throughout to make the concepts easier to follow. The book also contains over 200 exercises, many with solutions, making the book a useful resource for graduate students as well as researchers.

A Schanuel property for j

“…The dimension of a set B ⊆ X with respect to cl is the cardinality of any cl-independent set A ⊆ B such that cl(A) = B. For the basic properties of pregeometries, see Appendix C of [21].…”

The dynamical hierarchy for Roelcke precompact Polish groups

“…Let a ∈ M ω , n < ω; take p as in the hypothesis of the lemma, a ′ a realization of p. Given a formula f (x, y), the function df defined by df (b) = f (a ′ , b) is a <n -invariant. Since M is classical ℵ 0 -categorical, the structure M expanded with constants for the elements of a <n is ℵ 0 -categorical too (see [TZ12], Corollary 4.3.7). It follows that df (y) is an a <n -definable predicate, hence the conditions of Definition 4.1 are satisfied.…”

“…The unique countable dense linear order without endpoints, (Q, <), admits quantifier elimination (see [TZ12,§3.3.2]). This implies, for G = Aut(Q, <), that UC(G) is the closed unital algebra generated by the functions of the form g → (a = gb) and g → (a < gb) for elements a, b ∈ Q -where we think of the classical predicates x = y and x < y as {0, 1}-valued functions.…”

“…Let f = f (x, y) be an arbitrary formula and let a ∈ M α be a parameter. The cyclic G-space of f a (as defined after Remark 1.3) also has a name in the model-theoretic literature, at least for some authors: it coincides with the space of f -types over the orbit Ga as defined in [TZ12,p. 132].…”

Forcing a countable structure to belong to the ground model

“…Given a set A ⊆ C (whose size is, as usual, smaller than |C|) there is a class of formulas with parameters in C which are called the forking formulas over A. The precise definition can be found in, e.g., [15,Definition 7.1.7]. Given a tupleā and sets A ⊆ B ⊆ C, we writeā | A B when p = tp(ā/B) does not fork over A, meaning that no formula in p forks over A.…”