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Published online by Cambridge University Press: 26 June 2025
This article introduces some of the basic concepts and results from model theory, starting from scratch. The topics covered are be tailored to the model theory of fields and later articles. I will be using algebraically closed fields to illustrate most of these ideas. The tools described are quite basic; most of the material is due either to Alfred Tarski or Abraham Robinson. At the end I give some general references.
1. Languages and Structures
What is a mathematical structure? Some examples of mathematical structures we have in mind are the ordered additive group of integers, the complex field, and the ordered real field with exponentiation. To specify a structure we must specify the underlying set, some distinguished operations, some distinguished relations and some distinguished elements. For example, the ordered additive group of integers has underlying set ℤ and we distinguish the binary function +, the binary relation < and the identity element 0. For the ordered field of real numbers with exponentiation we have underlying set ℝ and might distinguish the binary functions + and x, the unary function exp, the binary relation < and the elements 0 and 1. Here is the formal definition.
DEFINITION 1.1. A structure M is given by the following data.
(i) A set M called the universe or underlying set of M.
(ii) A collection of functions where for some 1.
(iii) A collection of relations where for some.
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