deskriptivset

Deskriptivset, or descriptive set theory, is a branch of mathematical logic and set theory that studies the complexity and definability of sets and functions within Polish spaces—complete separable metric spaces such as the real line, Cantor space, and Baire space. The central concern is how sets can be constructed from simple sets using operations like countable unions, intersections, projections, and continuous images, and how these operations affect definability.

The theory organizes sets into hierarchies, notably the Borel hierarchy (Delta^0_alpha, Sigma^0_alpha, Pi^0_alpha) and the projective hierarchy (analytic Sigma^1_1, coanalytic Pi^1_1, and beyond). Analytic sets are continuous images of Borel sets; coanalytic sets are complements of analytic sets. Projective sets arise from iterated projections and complements applied to Borel sets. Researchers study regularity properties such as Lebesgue measurability, the Baire property, and the perfect set property, as well as the relationships between definability and these properties.

Classical results include Suslin's theorem, which characterizes Borel sets as those that are analytic and coanalytic, and the development of determinacy concepts tied to games, with consequences for regularity. The field also examines the Wadge hierarchy, reduction notions, and the role of large cardinals in proving consistency results.

Deskriptivset provides foundational tools for analysis, probability, topology, and model theory, and it serves as a bridge between pure set theory and concrete analysis. Key figures include Suslin, Luzin, Cantor, along with later contributors such as Kechris and Moschovakis.