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Handout from Today's Lecture MA532 Lecture Timothy Kohl Boston University April 23, 2020 Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 1 / 26 Cardinal Arithmetic Recall that one may define addition and multiplication of ordinals α = ot(A, A) β = ot(B, B ) α + β and α · β by constructing order relations on A ∪ B and B × A. For cardinal numbers the foundations are somewhat similar, but also somewhat simpler since one need not refer to orderings. Definition For sets A, B where |A| = α and |B| = β then α + β = |(A × {0}) ∪ (B × {1})|. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 2 / 26 The curious part of the definition is the two sets A × {0} and B × {1} which can be viewed as subsets of the direct product (A ∪ B) × {0, 1} which basically allows us to add |A| and |B|, in particular since, in the usual formula for the size of the union of two sets |A ∪ B| = |A| + |B| − |A ∩ B| which in this case is bypassed since, by construction, (A × {0}) ∩ (B × {1})= ∅ regardless of the nature of A ∩ B. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 3 / 26 Definition For sets A, B where |A| = α and |B| = β then α · β = |A × B|. One immediate consequence of these definitions is the following. Proposition If m, n are finite ordinals, then as cardinals one has |m| + |n| = |m + n|, (where the addition on the right is ordinal addition in ω) meaning that ordinal addition and cardinal addition agree. Proof. The simplest proof of this is to define a bijection f : (m × {0}) ∪ (n × {1}) → m + n by f (hr, 0i)= r for r ∈ m and f (hs, 1i)= m + s for s ∈ n. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 4 / 26 And in a similar fashion, one has the relatively obvious fact: Proposition If m, n are finite ordinals, then as cardinals |m| · |n| = |m · n|, where the multiplication on the right is ordinal multiplication in ω. Proof. Exercise. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 5 / 26 The addition and multiplication of cardinals differs from that ordinals. Proposition For cardinal numbers α,β, and γ: (a) α + β = β + α (b) α +0= α (c) (α + β)+ γ = α + (β + γ) (d) α · 0 = 0 (e) α · 1= α (f) α · 2= α + α (and similarly for α · n) (g) (α · β) = (β · α) (h) (α · β) · γ = α · (β · γ) (i) α · (β + γ)= α · β + α · γ Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 6 / 26 Proof. (a) There is a fairly obvious bijection f : (A × {0}) ∪ (B × {1}) → (B × {0}) × (A × {1}), namely ha, 0i 7→ ha, 1i and hb, 1i 7→ hb, 0i, and so |A| + |B| =c |B| + |A| (b) As ∅ × {1} = ∅ then (A × {0}) ∪ (∅ × {1}) = (A × {0}) ergo |A| + |∅| = |A|. (c) [Exercise] (d) For any set A we have A × ∅ = ∅ so |A| · |∅| = |∅|. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 7 / 26 The remaining statements are proved similarly. Proof. (e) There is a bijection A × 1= A × {0}→ A given by ha, 0i 7→ a which implies |A| · |1| = |A|. (f) As 2= {0, 1} there is yet another obvious bijection (A × {0}) ∪ (A × {1}) → A × 2= A × {0, 1} since, in fact, (A × {0}) ∪ (A × {1})= A × {0, 1} (g,h,i) Exercises. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 8 / 26 For the definition of ordinal exponentiation, we recall that for sets A and B, the functions f : B → A is denoted B A. Definition If α, β are cardinal numbers, then αβ denotes the unique cardinal γ such that, for sets A and B where |A| = α and |B| = β then γ = |B A|. For finite sets, for example, 32, a function f : 3 → 2 is a mapping from 3= {0, 1, 2} to 2= {0, 1}, i.e. f (0) = 0 or 1,f (1) = 0 or 1,f (2) = 0 or 1, ergo |32| = 23 = 8, where, not coincidentally |32| =c |P(3)|. And indeed, 2|A| = |P(A)| for any cardinality |A|. And of course, for finite cardinals m, n in general, the set nm has mn elements, i.e. cardinal exponentiation and ordinal exponentiation (in ω) are the same thing. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 9 / 26 We observe a number of facts about the exponentiation of cardinal numbers. Proposition For cardinals α, β, and γ: (a) α0 = 1 (b) α1 = α (c) α2 = α · α (d) αn = αn−1 · α for n a finite cardinal (e) if β 6= 0, 0β = 0 (f) 1β = 1 (g) αβ+γ = αβ · αγ (h) αβ·γ = (αβ)γ (i) (α · β)γ = αβ · βγ Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 10 / 26 Proof. (a) This is somewhat curious statement since α0 represents the cardinality of ∅α, the totality of functions from ∅ to α, but this consists of the empty set so it equals {∅} = {0} = 1. (b) Each function in 1α has domain consisting of a single element, and so the number of functions is in bijective correspondence with the cardinality of α, ergo α1 = α. (c,d) Exercises. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 11 / 26 Here are the proofs of the remaining statements. Proof. (e) In contrast with α0, for 0β one is looking at the set of function B → ∅ for some set B with α = |B| then we must realize that no function with a non-empty domain can have range ∅. (f) Exercise (g) For sets A, B, C, with cardinal numbers α, β and γ one compares functions (B × {0}) ∪ (C × {1}) → A with the direct product (B A) × (C A) and realize that since (B × {0}) ∩ (C × {1})= ∅ any f : B × {0}→ A and g : C × {1}→ A are ’independent’ and so determine a unique function hf , gi∈ (B A) × (C A). (h,i) Exercises. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 12 / 26 Cofinality For the definition we give next, we need to give a definition of limit. Definition If α> 0 is a limit ordinal, and hγζ | ζ < αi be a non-decreasing sequence of ordinals (i.e. ζ < η implies γζ ≤ γη) then the limit of the sequence lim γζ = sup{γζ | ζ < α} ζ→α and a sequence of ordinals hγα | α ∈ Oni is normal if it is increasing and continuous, i.e. for every limit ordinal α, γα = lim γζ ζ→α Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 13 / 26 Definition If α> 0 is a limit ordinal, we say that an increasing sequence hαζ | ζ < βi, β a limit ordinal, is cofinal in α if lim αζ = α. ζ→β Similarly, A ⊆ α is a cofinal in α if supA = α. If α is an infinite limit ordinal, the confinality of α is cf (α) = the least limit ordinal β such that there is an increasing sequence hαζ | ζ < βi with lim αζ = α ζ→β Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 14 / 26 A slightly easier definition perhaps is this: Definition Let α, β be ordinals and let f : α → β be a function, f is cofinal (or α cofinally (in)to β) if for all γ < β, there exists δ < α such that γ ≤ f (δ). Let β ∈ On then the confinality of β, denoted cf (β), is the minimal α such that there is a cofinal function from α to β. So cf (β) ≤ β and is a cardinal for any β, and if β is a successor ordinal (i.e. not a limit ordinal) then cf (β) = 1 via the function f (0) = S β. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 15 / 26 A more familiar way of viewing cofinality is from the theory of sequences from real analysis/calculus. In particular, if {an} is a sequence, and if {kn} is defined as a sequence of indices such that lim kn = ∞ then {akn } is a subsequence of {an}. n→∞ One of the canonical properties of convergent sequences {an} is that if L = lim an (finite) then for any subsequence {akn } one has lim akn = L n→∞ n→∞ as well. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 16 / 26 1 1 For example, given the sequence, { 2n } we have that { 4n } is a subsequence, and both tend to zero as n →∞. Although it’s not essential here, we observe that there exists sequences {an} with subsequences {bn} and {cn} such that lim bn = L1 and n→∞ lim cn = L2 (where L1 and L2 are finite) but where L1 6= L2. n→∞ In this situation we conclude that lim an does not exist, so the n→∞ convergence of the sequence implying the convergence of all subsequences is a pretty stringent condition. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 17 / 26 And, in terms of sets, the cofinality can be be viewed as a kind of ’density’ 1 1 of one in the other, for example with the { 2n } and { 4n } sequences earlier, as sets 1 1 1 1 { } = { , , ,... } 4n 4 16 64 is cofinal in 1 1 1 1 1 1 1 1 { } = { , , , , , , ,... } 2n 2 4 8 16 32 64 128 1 1 since every other term of { 2n } is one of the elements of { 4n }. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 18 / 26 So for a given ordinal α the cofinality is basically the order type of any index set of a sequence whose terms comprise a cofinal set.
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