9.2 Large Cardinals
An introduction to large cardinal axioms, inaccessible cardinals, measurable cardinals, elementary embeddings, and consistency strength.
Tag
Set Theory
20 notes
9.1 Forcing
An introduction to forcing, generic filters, forcing names, the forcing relation, and the basic extension theorem.
8.3 Constructible Universe (L)
The constructible universe, definable subsets, the hierarchy L_alpha, and the axiom of constructibility.
8.2 Equivalent Formulations
Equivalent forms of the axiom of choice, including Zorn's lemma, the well ordering theorem, maximal principles, and right inverses of surjections.
8.1 Axiom of Choice
The axiom of choice, choice functions, indexed families, and first consequences in axiomatic set theory.
7.4 Arithmetic of Cardinals
Cardinal addition, multiplication, exponentiation, finite and infinite cardinal arithmetic, and basic comparison laws.
7.1 Sets, Relations, Functions
Basic set theoretic language, including sets, membership, subsets, operations, relations, equivalence relations, order relations, and functions.
9.5 Applications in Analysis and Topology
Applications of advanced set theory to analysis and topology, including regularity properties, Banach spaces, measure theory, and topological classification.
9.4 Determinacy Principles
An introduction to infinite games, determined games, the axiom of determinacy, projective determinacy, and consequences for sets of reals.
9.3 Descriptive Set Theory
An introduction to Polish spaces, Borel sets, analytic sets, projective sets, regularity properties, and the role of definability in set theory.
Chapter 9. Advanced Set Theory
Forcing, large cardinals, descriptive set theory, determinacy principles, and applications in analysis and topology.
8.5 Independence Phenomena
Independence results in set theory, including the axiom of choice and the continuum hypothesis, and the methods used to establish independence.
8.4 Consistency Results
Relative consistency, inner models, constructibility, and the role of consistency results in axiomatic set theory.
Chapter 8. Axiomatic Systems
Axiom of Choice, equivalent formulations, constructible universe, consistency results, and independence phenomena.
7.5 Zermelo Fraenkel Axioms (ZF, ZFC)
The axioms of Zermelo Fraenkel set theory, the role of choice, and the use of axioms as a foundation for mathematics.
7.3 Ordinals and Well Ordering
Well ordered sets, order isomorphisms, ordinals, successor ordinals, limit ordinals, and transfinite induction.
7.2 Cardinality and Countability
Cardinality, finite and infinite sets, countable sets, uncountable sets, and Cantor diagonal arguments.
Chapter 7. Basic Set Theory
Basic set theoretic notions including sets, relations, functions, cardinality, ordinals, well ordering, cardinal arithmetic, and the ZF and ZFC axioms.
1.2 Sets, Types, and Universes
Three ways to organize a domain of discourse for mathematics: sets, types, and universes — and how they relate.
03. Logic and Foundations
Formal logic, set theory, computability, and the foundations of mathematics treated as a formal system.