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Axiomatic Set Theory Alexandru Baltag (ILLC, University of Amsterdam) LECTURE NOTES 5: AC, the Well-Ordering Theorem and Operations on Ordinals 1 Various Formulations of the Axiom of Choice Axiom of Choice (AC) (first formulation): If F is a set of pairwise-disjoint, non-empty sets, then there exists a set M (called a “choice set”) that consists of exactly one element from each member of F. (AC’) (second formulation): If F is a set of non-empty sets, then there exists a function S f : F → F (called a “choice function”) such that f (X) ∈ X for all X ∈ F. (AC”) (third formulation): If F is a set of non-empty sets, then the Cartesian product ΠF = ΠX∈F X is non-empty. Proposition (in ZF): (AC) ⇔ (AC 0 ) ⇔ (AC 00 ). 2 The Well-Ordering Principle Another equivalent formulation is Zermelo’s Well-Ordering Principle (WO), also known as the Well-Ordering Theorem: (WO) Well-Ordering Theorem (in ZF C): Every set can be well-ordered. Proposition (in ZF , see Theorem 2.7.3 in Devlin): (AC) ⇔ (W O). 3 Zorn’s Lemma Another equivalent formulation is “Zorn’s lemma”. An element a of a poset (P, ≤) is maximal (in P ) iff there is no element b ∈ P s.t. a < b. A chain in a poset (P, ≤) is a subset of P that is totally ordered (by ≤). An element a of a poset (P, ≤) is an upper bound of a subset B ⊆ P iff b ≤ a for all b ∈ B. “Zorn’s Lemma” (ZL): If every chain in a poset (P, ≤) has an upper bound (in P ), then P has a maximal element. Proposition (in ZF , see Theorem 2.7.5 in Devlin): (AC) ⇔ (ZL). 4 Ordinals in ZF C DEF: A set X is transitive iff ∀x∀y(y ∈ x ∧ x ∈ X ⇒ y ∈ X) Lemma 3.1.1. (Devlin): A set X is an ordinal iff it is transitive and totally ordered by ∈. Consequences (provable in ZF , see Lemmas 3.1.2 and 3.1.3 in Devlin): If α is an ordinal, then its immediate successor α + 1 := α ∪ {α} is an ordinal. S If A is a set of ordinals, then A is an ordinal. 5 Proving existence of ordinals in ZF The above results allow us to prove in ZF the existence of all finite ordinals, and then (using the Axiom of Infinity and the Replacement Axiom) the existence of ω. Next, one can iterate the successor function to generate all ordinal ω + n. To show existence of ω + ω, let F : ω → V, F (n) = ω + n By Replacement, f [ω] = {f (n)|n ∈ ω} is a set (of ordinals). S By the above result, f [ω] = ω + ω is an ordinal. Similarly, one can show existence of ω + ω + ω etc. 6 Addition of Ordinals To define ordinal sum α + β, we take the (unique ordinal isomorphic to the) disjoint union of the two ordinals well-ordered it by putting first the elements of α (in their usual order as ordinals) and then the elements of β (in their usual order). Formally: take the set A = (α × {0}) ∪ (β × {1}) with the anti-lexicographic well-ordering (µ, i) <A (τ, j) ⇔ (i < j) ∨ (i = j ∧ µ < τ ), then define α + β := Ord(A, <A ) where Ord(A, <A ) is the unique ordinal isomorphic to (A, <A ). 7 Properties of Ordinal Addition Proposition: Ordinal addition is associative: α + (β + γ) = (α + β) + γ. BUT it is NOT commutative! n+ω =ω but ω+n>ω 8 Monotonic Laws for Ordinal Addition The First Monotonic Law for Addition: Ordinal addition is strictly increasing in the second argument: α<β ⇒ γ+α<γ+β The Second Monotonic Law for Addition: Ordinal addition is monotonic in the first argument: α≤β ⇒ α+γ ≤β+γ BUT ordinal addition is NOT strictly increasing in the first argument: 1 < 2, but 1 + ω = ω = 2 + ω. 9 Ordinal Subtraction Theorem: If α ≥ β are ordinals, then there exists a unique ordinal γ (called the difference of the two ordinals, and denoted by α − β) such that α = β + γ. CONSEQUENCE: if α ≥ β, then α = β + (α − β). EXAMPLES: ω−n=ω ω2 − ω = ω2 10 Monotonic Laws for Ordinal Subtraction Difference is strictly increasing in the first argument: α>β ≥γ ⇒ α−γ >β−γ Difference is anti-monotonic in the second argument: γ ≥α≥β ⇒ γ−α≤γ−β BUT it is NOT strictly decreasing in the second argument: ω > 2 > 1 but ω − 2 = ω = ω − 1. 11 Ordinal Multiplication To define the ordinal multiplication α · β of two ordinals α and β, we take (the unique ordinal isomorphic to) the Cartesian Product α×β well-ordered with the anti-lexicographic order (µ, η) < (µ0 , η 0 ) ⇔ (η < η 0 ) ∨ (η = η 0 ∧ µ < µ0 ), (for µ ∈ α, η ∈ β). As we’ll see, ordinal multiplication can also be seen as iterated ordinal addition. 12 Ordinal Addition of A Sequence of Ordinals We generalize the above ordinal sum to well-ordered sequences (αη )η<λ of ordinals. (i.e. functions f : λ → On, with f (η) = αη ). The ordinal sum X αη η<λ is defined by taking the disjoint union [ (αη × {η}) A= η<λ together with the anti-lexicographic well-ordering 0 0 0 0 0 (µ, η) < (µ , η ) ⇔ (η < η ) ∨ (η = η ∧ µ < µ ), X and putting αη := Ord(A, <A ). η<λ 13 Ordinal Multiplication Then it is easy to see that ordinal multiplication is iterated addition: X α·β = α η<β (Formally, this is the ordinal sum of the sequence given by the function f : β → On, given by f (η) = α, for every η ∈ β ). 14 Properties of Ordinal Multiplication Ordinal Multiplication is associative α · (β · γ) = (α · β) · γ and distributes to the right over addition: α · (β + γ) = α · β + α · γ But it is NOT commutative n · ω = ω, but ω · n > ω and it does NOT distribute to the left: (1 + 1) · ω = 2 · ω = ω but 1 · ω + 1 · ω = ω + ω > ω. 15 Monotonic Laws for Ordinal Multiplication The First Monotonic Law for Multiplication: Ordinal multiplication with a non-zero first argument is strictly increasing in the second argument: α < β ∧ γ 6= 0 ⇒ γ · α < γ · β The Second Monotonic Law for Multiplication: Ordinal multiplication is monotonic in the first argument: α≤β ⇒ α·γ ≤β·γ BUT ordinal multiplication (even with a non-zero second argument) is NOT strictly increasing in the first argument: 1 < 2, but 1 · ω = ω = 2 · ω. 16 Distributivity to the Right over Subtraction Ordinal multiplication is distributive to the right with respect to ordinal subtraction: α · (β − γ) = α · β − α · γ But it does NOT distribute to the left: (2 − 1) · ω = 1 · ω = ω while 2·ω−1·ω =ω−ω =0 17 Division with Remainder Theorem (Division with Remainder): If α is an ordinal and β > 0 is a non-zero ordinal, then there exist uniquely determined ordinals γ and δ, such that α = β · γ + δ, and δ < β. 18 Limit of a Sequence of Ordinals For a given limit ordinal λ, let (αη )η<λ be a λ-sequence of ordinals (i.e. a function f : λ → On, with f (η) = αη ). For an ordinal α, we write α = lim αη η<λ iff ∀β < α ∃η < λ ∀ζ (η < ζ < λ ⇒ β < αζ ≤ α) . It is easy to see that, if α exists, then it is unique. If it exists, we call α the limit of the sequence (αη )η<λ . Lemma 3.4.1 (Devlin): For every limit ordinal λ, every increasing λ-sequence (αη )η<λ has a (unique) limit, given by [ lim αη = αη . η<λ η<λ 19 Properties of limits lim (α + βη ) = α + lim βη η<λ η<λ But NOT the other way around: e.g. lim (n + ω) = lim ω = ω, BUT lim n + ω = ω + ω > ω. n<ω n<ω n<ω Similarly: lim (α · βη ) = α · lim βη η<λ η<λ But NOT the other way around: e.g. 2 lim (n · ω) = lim ω = ω, BUT lim n · ω = ω · ω = ω > ω. n<ω n<ω n<ω 20 Ordinal Exponentiation We can define ordinal exponention by recursion on ordinals: α0 = 1 αβ+1 = αβ · α αβ = lim αγ , if β is a limit ordinal. γ<β (More precisely, for each given ordinal α, we use the above recursive clauses to define a class function Fα : On → On, Fα (β) = αβ .) 21 Properties of Ordinal Exponentiation Lemma 3.5.1 (Devlin): αβ · αγ = αβ+γ β γ α = αβ·γ But left-distributivity (of exponentiation over multiplication) FAILS: (ω · 2)2 6= ω 2 · 22 Other properties: 1α = 1 α1 = α 0α = 0, for α > 0 nω = ω for any 1 < n < ω 22 Monotonic Laws for Ordinal Exponentiation The First Monotonic Law for Exponentiation: Ordinal exponentiation with a non-zero first argument is strictly increasing in the second argument: α < β ∧ γ 6= 0 ⇒ γ α < γ β The Second Monotonic Law for Exponentiation: Ordinal exponentiation is monotonic in the first argument: α ≤ β ⇒ αγ ≤ β γ BUT ordinal exponentiation (even with a non-zero second argument) is NOT strictly increasing in the first argument: 2 < 3, but 2ω = ω = 3ω . 23