The Continuum Hypothesis, the Axiom of Choice, and Lebesgue Measurability Chris Lambie-Hanson CMU Graduate Student Seminar 11 October 2011 1 For all x 2 S, x ≤ x. 2 For all x; y 2 S, if x ≤ y and y ≤ x, then x = y. 3 For all x; y; z 2 S, if x ≤ y and y ≤ z, then x ≤ z. 4 For all x; y 2 S, x ≤ y or y ≤ x. A linear order ≤ on a set S is a well-order if, for every nonempty X ⊆ S, there is a ≤-least element in X , i.e. there is x 2 X such that, for all y 2 X , x ≤ y. Definitions If S is a set, then a linear order on S is a binary relation ≤ such that 2 For all x; y 2 S, if x ≤ y and y ≤ x, then x = y. 3 For all x; y; z 2 S, if x ≤ y and y ≤ z, then x ≤ z. 4 For all x; y 2 S, x ≤ y or y ≤ x. A linear order ≤ on a set S is a well-order if, for every nonempty X ⊆ S, there is a ≤-least element in X , i.e. there is x 2 X such that, for all y 2 X , x ≤ y. Definitions If S is a set, then a linear order on S is a binary relation ≤ such that 1 For all x 2 S, x ≤ x. 3 For all x; y; z 2 S, if x ≤ y and y ≤ z, then x ≤ z. 4 For all x; y 2 S, x ≤ y or y ≤ x. A linear order ≤ on a set S is a well-order if, for every nonempty X ⊆ S, there is a ≤-least element in X , i.e. there is x 2 X such that, for all y 2 X , x ≤ y. Definitions If S is a set, then a linear order on S is a binary relation ≤ such that 1 For all x 2 S, x ≤ x. 2 For all x; y 2 S, if x ≤ y and y ≤ x, then x = y. 4 For all x; y 2 S, x ≤ y or y ≤ x. A linear order ≤ on a set S is a well-order if, for every nonempty X ⊆ S, there is a ≤-least element in X , i.e. there is x 2 X such that, for all y 2 X , x ≤ y. Definitions If S is a set, then a linear order on S is a binary relation ≤ such that 1 For all x 2 S, x ≤ x. 2 For all x; y 2 S, if x ≤ y and y ≤ x, then x = y. 3 For all x; y; z 2 S, if x ≤ y and y ≤ z, then x ≤ z. A linear order ≤ on a set S is a well-order if, for every nonempty X ⊆ S, there is a ≤-least element in X , i.e. there is x 2 X such that, for all y 2 X , x ≤ y. Definitions If S is a set, then a linear order on S is a binary relation ≤ such that 1 For all x 2 S, x ≤ x. 2 For all x; y 2 S, if x ≤ y and y ≤ x, then x = y. 3 For all x; y; z 2 S, if x ≤ y and y ≤ z, then x ≤ z. 4 For all x; y 2 S, x ≤ y or y ≤ x. Definitions If S is a set, then a linear order on S is a binary relation ≤ such that 1 For all x 2 S, x ≤ x. 2 For all x; y 2 S, if x ≤ y and y ≤ x, then x = y. 3 For all x; y; z 2 S, if x ≤ y and y ≤ z, then x ≤ z. 4 For all x; y 2 S, x ≤ y or y ≤ x. A linear order ≤ on a set S is a well-order if, for every nonempty X ⊆ S, there is a ≤-least element in X , i.e. there is x 2 X such that, for all y 2 X , x ≤ y. The Axiom of Choice is equivalent to the Well-ordering Theorem, which asserts that every set can be well-ordered. The Axiom of Choice The Axiom of Choice is the assertion: For every family of nonempty sets F, there is a function g such that dom(g) = F and, for every X 2 F, g(X ) 2 X . The Axiom of Choice The Axiom of Choice is the assertion: For every family of nonempty sets F, there is a function g such that dom(g) = F and, for every X 2 F, g(X ) 2 X . The Axiom of Choice is equivalent to the Well-ordering Theorem, which asserts that every set can be well-ordered. • A natural number n is an ordinal describing a well-ordering with n elements. • The ordinal describing the order type of the natural numbers is !. • The ordinal describing the order type of the natural numbers plus one element larger than all of the natural numbers is ! + 1. An ordinal α is actually a set of ordinals, well-ordered by 2, of order-type α. If α = β + 1 = β [ fβg, then α is called a successor ordinal. Otherwise, it is called a limit ordinal. Ordinals, Informally An ordinal number describes the order type of a well-ordering. • The ordinal describing the order type of the natural numbers is !. • The ordinal describing the order type of the natural numbers plus one element larger than all of the natural numbers is ! + 1. An ordinal α is actually a set of ordinals, well-ordered by 2, of order-type α. If α = β + 1 = β [ fβg, then α is called a successor ordinal. Otherwise, it is called a limit ordinal. Ordinals, Informally An ordinal number describes the order type of a well-ordering. • A natural number n is an ordinal describing a well-ordering with n elements. • The ordinal describing the order type of the natural numbers plus one element larger than all of the natural numbers is ! + 1. An ordinal α is actually a set of ordinals, well-ordered by 2, of order-type α. If α = β + 1 = β [ fβg, then α is called a successor ordinal. Otherwise, it is called a limit ordinal. Ordinals, Informally An ordinal number describes the order type of a well-ordering. • A natural number n is an ordinal describing a well-ordering with n elements. • The ordinal describing the order type of the natural numbers is !. An ordinal α is actually a set of ordinals, well-ordered by 2, of order-type α. If α = β + 1 = β [ fβg, then α is called a successor ordinal. Otherwise, it is called a limit ordinal. Ordinals, Informally An ordinal number describes the order type of a well-ordering. • A natural number n is an ordinal describing a well-ordering with n elements. • The ordinal describing the order type of the natural numbers is !. • The ordinal describing the order type of the natural numbers plus one element larger than all of the natural numbers is ! + 1. If α = β + 1 = β [ fβg, then α is called a successor ordinal. Otherwise, it is called a limit ordinal. Ordinals, Informally An ordinal number describes the order type of a well-ordering. • A natural number n is an ordinal describing a well-ordering with n elements. • The ordinal describing the order type of the natural numbers is !. • The ordinal describing the order type of the natural numbers plus one element larger than all of the natural numbers is ! + 1. An ordinal α is actually a set of ordinals, well-ordered by 2, of order-type α. Ordinals, Informally An ordinal number describes the order type of a well-ordering. • A natural number n is an ordinal describing a well-ordering with n elements. • The ordinal describing the order type of the natural numbers is !. • The ordinal describing the order type of the natural numbers plus one element larger than all of the natural numbers is ! + 1. An ordinal α is actually a set of ordinals, well-ordered by 2, of order-type α. If α = β + 1 = β [ fβg, then α is called a successor ordinal. Otherwise, it is called a limit ordinal. The cardinality of A, written jAj, can be thought of as the equivalence class of all sets for which bijections with A exist. For two sets A and B, we say jAj jBj if there is an injective function f : A ! B. Under the Axiom of Choice, a cardinality is often identified with the smallest ordinal of that cardinality. Thus, under AC, cardinalities are well-ordered by . If α is an ordinal, the α-th infinite cardinal is written @α. The smallest ordinal of cardinality @α is written !α. Cardinals Two sets A and B have the same cardinality if there is a bijection f : A ! B. For two sets A and B, we say jAj jBj if there is an injective function f : A ! B. Under the Axiom of Choice, a cardinality is often identified with the smallest ordinal of that cardinality. Thus, under AC, cardinalities are well-ordered by . If α is an ordinal, the α-th infinite cardinal is written @α. The smallest ordinal of cardinality @α is written !α. Cardinals Two sets A and B have the same cardinality if there is a bijection f : A ! B. The cardinality of A, written jAj, can be thought of as the equivalence class of all sets for which bijections with A exist. Under the Axiom of Choice, a cardinality is often identified with the smallest ordinal of that cardinality.