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Maths › 03. Logic and Foundations ›

9.3 Descriptive Set Theory

Descriptive set theory studies definable subsets of spaces such as the real line, the Baire space, and the Cantor space, and its main goal is to understand how the complexity of a set is related to the way the set can be described. The subject begins with topological spaces that are rich enough to support analysis but structured enough to allow precise classification, and the central examples are spaces whose...
logic set theory descriptive set theory definability May 2, 2026
Maths › 03. Logic and Foundations ›

6.1 Definable Sets

Definability is one of the central ideas of model theory. Once a first order language and a structure have been fixed, formulas can be used to describe subsets of the domain, relations on the domain, and functions between definable sets. In this way, syntax gives rise to geometry inside a structure: a formula becomes a description of a set of points satisfying a condition. In this section, we work with...
logic model theory definability first order logic May 2, 2026
Maths › 03. Logic and Foundations ›

6. Definability and Types

Definability studies which sets, relations, and functions can be described inside a structure using formulas of a fixed language, and it is one of the main ways model theory connects syntax with the internal geometry of mathematical structures. The chapter begins with definable sets and functions, where a formula with free variables is used to specify a collection of tuples from a model, and this gives a precise way to...
logic model theory definability types May 2, 2026