Model Theory for Algebra and Algebraic Geometry

In mathematical logic, we use first-order languages to describe mathematical structures. Intuitively, a structure is a set that we wish to study equipped with a collection of distinguished functions, relations, and elements. We then choose a language where we can talk about the distinguished functions, relations, and elements and nothing more. For example, when we study the ordered field of real numbers with the exponential function, we study the structure (R,+, ·, exp, <, 0, 1), where the underlying set is the set of real numbers, and we distinguish the binary functions addition and multiplication, the unary function x 7→ e, the binary order relation, and the real numbers 0 and 1. To describe this structure, we would use a language where we have symbols for +, ·, exp, < , 0, 1 and can write statements such as ∀x∀y exp(x) · exp(y) = exp(x + y) and ∀x (x > 0 → ∃y exp(y) = x). We interpret these statements as the assertions “ee = e for all x and y” and “for all positive x, there is a y such that e = x.” For another example, we might consider the structure (N,+, 0, 1) of the natural numbers with addition and distinguished elements 0 and 1. The natural language for studying this structure is the language where we have a binary function symbol for addition and constant symbols for 0 and 1. We would write sentences such as ∀x∃y (x = y + y ∨ x = y + y + 1), which we interpret as the assertion that “every number is either even or 1 plus an even number.”