Map of Set Theory Set: A collection of distinct objects. - Finite set: A set with a finite number of elements. - Infinite set: A set with an infinite number of elements. - Empty set (null set): A set with no elements, denoted by {} or ∅. - Singleton set: A set with exactly one element. Subset: A set A is a subset of set B (A ⊆ B) if every element of A is also an element of B. - Proper subset: A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B. Superset: B is a superset of A if A ⊆ B. Power set: The set of all subsets of a given set A, denoted by P(A) or 2^A. Cardinality: The number of elements in a set. - |A|: The cardinality of set A. - Countable set: A set with the same cardinality as the natural numbers (ℵ₀). - Uncountable set: A set with a cardinality greater than ℵ₀, such as the real numbers. Set operations: - Union (A ∪ B): The set containing all elements that are in A, B, or both. - Intersection (A ∩ B): The set containing all elements that are in both A and B. - Difference (A \ B): The set containing all elements of A that are not in B. - Symmetric difference (A △ B): The set containing elements that are in either A or B, but not both. - Cartesian product (A × B): The set of all ordered pairs (a, b) where a ∈ A and b ∈ B. Complement: The complement of set A (A') is the set of all elements in the universal set that are not in A. Venn diagrams: Graphical representations of sets and their relationships. Set identities: - Commutative laws: A ∪ B = B ∪ A, A ∩ B = B ∩ A - Associative laws: (A ∪ B) ∪ C = A ∪ (B ∪ C), (A ∩ B) ∩ C = A ∩ (B ∩ C) - Distributive laws: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C), A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) - Identity laws: A ∪ ∅ = A, A ∩ U = A (U is the universal set) - Complement laws: A ∪ A' = U, A ∩ A' = ∅ - Idempotent laws: A ∪ A = A, A ∩ A = A - Absorption laws: A ∪ (A ∩ B) = A, A ∩ (A ∪ B) = A - De Morgan's laws: (A ∪ B)' = A' ∩ B', (A ∩ B)' = A' ∪ B' Set-builder notation: {x | P(x)}, which reads "the set of all x such that P(x) is true." Axioms of set theory: - Axiom of extensionality: Two sets are equal if and only if they have the same elements. - Axiom of regularity: Every non-empty set A contains an element that is disjoint from A. - Axiom schema of specification: {x ∈ A | P(x)} is a set for any set A and predicate P. - Axiom of pairing: For any a and b, there exists a set {a, b} that contains exactly a and b. - Axiom of union: For any set of sets A, there is a set containing all elements that belong to at least one set in A. - Axiom schema of replacement: If F is a function, then for any set A, the image of A under F is also a set. - Axiom of infinity: There exists an infinite set. - Axiom of power set: For any set A, there exists a set containing all subsets of A. - Well-ordering theorem: Every set can be well-ordered. Ordinal numbers: Generalization of natural numbers used to describe the order type of well-ordered sets. - Successor ordinal: For an ordinal α, the successor ordinal is α + 1. - Limit ordinal: An ordinal that is not zero or a successor ordinal. - Transfinite induction: A proof technique for properties of ordinal numbers. Cardinal numbers: Generalization of natural numbers used to describe the cardinality of sets. - Aleph numbers (ℵ₀, ℵ₁, ...): Infinite cardinal numbers. - Beth numbers (ℶ₀, ℶ₁, ...): Cardinal numbers defined by the beth function, where ℶ₀ = ℵ₀ and ℶ₀ = 2^ℶ₀. - Continuum hypothesis: There is no set with a cardinality between ℵ₀ and 2^ℵ₀ (the cardinality of the real numbers). Zermelo-Fraenkel set theory (ZF): A commonly used axiomatic system for set theory. - Zermelo-Fraenkel set theory with the axiom of choice (ZFC): ZF with the addition of the axiom of choice. Axiom of choice: For any set A of non-empty sets, there exists a function f such that f(X) ∈ X for all X ∈ A. - Equivalent statements: Zorn's lemma, well-ordering theorem, Tychonoff's theorem. Constructible universe (L): A class of sets that can be constructed using a specific transfinite recursion process. - Axiom of constructibility (V=L): The statement that every set is constructible. Forcing: A technique for extending models of set theory to create new models with desired properties. - Cohen forcing: A type of forcing used to prove the independence of the continuum hypothesis from ZFC. Large cardinals: Cardinal numbers with properties so strong that their existence cannot be proved in ZFC. - Inaccessible cardinal: A cardinal κ that is uncountable, regular, and a strong limit cardinal. - Measurable cardinal: A cardinal κ for which there exists a κ-complete non-principal ultrafilter on κ. - Supercompact cardinal: A cardinal κ such that for all λ ≥ κ, there is an elementary embedding from V into a transitive class M with critical point κ such that λ < j(κ) and Mλ ⊆ M. Inner model theory: The study of canonical inner models of set theory, such as L and its generalizations. - Constructible hierarchy: The transfinite sequence of sets (Lα | α ∈ On) defined by transfinite recursion. - Gödel's constructible universe: The class L = ⋃(α ∈ On) Lα. - Jensen's covering lemma: A statement about the relationship between L and V under the assumption of 0^#. Descriptive set theory: The study of definable subsets of Polish spaces (complete, separable metric spaces). - Borel sets: Sets that can be constructed from open sets using countable unions, countable intersections, and complements. - Analytic sets: Continuous images of Borel sets. - Projective hierarchy: A hierarchy of sets defined by alternating projections and complementation, starting with analytic sets. Determinacy: The study of two-player games with perfect information. - Gale-Stewart games: Infinite games where players alternate choosing elements from a set X to form an infinite sequence, and the winner is determined by a payoff set A ⊆ X^ω. - Axiom of determinacy (AD): The statement that for every A ⊆ ω^ω, the Gale-Stewart game with payoff set A is determined (i.e., one of the players has a winning strategy). - Projective determinacy: The statement that every projective set is determined. Infinite combinatorics: The study of combinatorial properties of infinite sets. - Ramsey's theorem: For any positive integers n and k, there exists a positive integer R(n, k) such that any k-coloring of the n-element subsets of a set with at least R(n, k) elements contains a monochromatic subset of size n. - Erdős-Rado theorem: A generalization of Ramsey's theorem to infinite sets. - Partition calculus: The study of generalizations and variants of Ramsey's theorem. Set-theoretic topology: The study of topological properties in the context of set theory. - Ultrafilters: Maximal filters on a set. - Stone-Čech compactification: A universal compactification of a topological space. - Martin's axiom: A statement about the existence of generic filters for certain partial orders.