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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 ›

9.4 Determinacy

Determinacy principles study infinite games between two players, and they assert that for certain classes of games one of the two players has a winning strategy. These principles are important in set theory because they give strong regularity properties for definable sets of reals and provide a deep alternative perspective to the axiom of choice. Infinite Games We begin with games of perfect information played by two players, usually called...
logic set theory determinacy descriptive set theory May 2, 2026
Maths › 03. Logic and Foundations ›

9. Advanced Set Theory

Advanced set theory studies methods and principles that go beyond the basic axioms of ZF and ZFC, especially when ordinary axioms do not decide important mathematical questions. The chapter begins with forcing, which is a method for constructing new models of set theory from old ones, and it provides the main technique for proving independence results such as the independence of the continuum hypothesis. Large cardinals are then introduced as...
logic set theory forcing large cardinals descriptive set theory May 2, 2026