Math 455 Some notes on Cardinality and Transfinite Induction (David Ross, UH-Manoa Dept. of Mathematics) 1 Cardinality Recall the following notions: function, relation, one-to-one, onto, on-to-one correspondence, injection, surjection, bijection, equivalence relation, An, AB =the set of all functions from A into B Identify a natural number with the set of its predecessors, n = f0; 1; : : : ; n−1g. Today: 0 = fg 2 N Definition 1.1 A is equinumerous with B provided there is a bijection from A onto B. Other, equivalent notation/terminology for \A is equinumerous with B": a. A and B have the same cardinality b. A ≈ B c. card(A) = card(B) We seem to be referring to a concept called \cardinality" without really defining it; this is especially true for (c), where the notation looks like we are setting two functions equal to one another. Examples: 2 >0 N ≈ Z ≈ N ≈ Q ≈ Q 2 R ≈ P(N) ≈ R ≈ ff : R ! R : f is continuousg 1 The following easy theorem proves that ≈ is \like" an equivalence relation. Theorem 1.1 For any sets A; B; and C: 1. A ≈ A 2. A ≈ B =) B ≈ A 3. A ≈ B &B ≈ C =) A ≈ C It will be convenient to have a slightly more general notation: Definition 1.2 A has no greater cardinality than B (or card(A) ≤ card(B)) provided there is an injection from A into B. Theorem 1.2 For any nonempty sets A and B the following are equivalent: 1. card(A) ≤ card(B) 2. For some C ⊆ B; card(A) = card(C) 3. There is a function g from B onto A. Remark: What if A = ;? Theorem 1.3 (Cantor-Schroder-Bernstein) If card(A) ≤ card(B) and card(B) ≤ card(A) then card(A) = card(B) Theorem 1.4 (Cantor) For any set A, card(A) 6= card(P (A)) (in fact, card(A) < card(P (A)) Proof: Let g be any 1 − 1 function from A into P (A). Put B = fx 2 A : x2 = g(x)g Claim: B2 = range(g). To see this, let x0 2 A, and consider 2 cases: (i) x0 2 g(x0); then x0 2= B, so B 6= g(x0). (ii) x0 2= g(x0); then x0 2 B, so B 6= g(x0). Either way, B 6= g(x0). Since x0 was arbitrary in A, B2 = range(g). Since g was arbitrary, A 6≈ P (A). (Note that a 7! fag is an injection from A into P (A), so card(A) ≤ card(P (A)).) Theorem 1.5 (AC) Any sets A and B are comparable; that is, card(A) < card(B); card(A) = card(B); or card(A) > card(B) This is sometimes used to define the Axiom of Choice (AC). Proposition 1.1 For every set A, P (A) ≈ A2 = the set of functions from A into f0; 1g Proof: Identify every subset of A with its characteristic function. Cor 1.1 N 6≈ R 2 1.1 Finite sets Definition 1.3 • A is finite if 9m 2 N A ≈ m, A is not finite otherwise • A is infinite if card(N) ≤ card(A), A is not infinite otherwise • A is Dedekind-infinite if 9B ( AA ≈ B, Dedekind-finite otherwise Theorem 1.6 Suppose A 6= ;. The following are equivalent: 1. A is finite 2. A is not infinite 3. A is Dedekind-finite 4. There exists an injection f : A ! m for some m 2 N 5. There exists a surjection g : m ! A for some m 2 N Cor 1.2 If A is finite then there is a unique m 2 N with A ≈ m (The proof of Finite,Dedekind Finite is surprisingly hard!) Remark: For A finite we can now define card(A) to be the unique natural number m such that A ≈ m. 1.2 Countability Definition 1.4 A is countable (or enumerable) if either A is finite or A ≈ N. If A is any countable and infinite set then we write card(A) = @0, and we write card(A) ≤ @0 to mean that card(A) ≤ card(N). Remark: We will call a set A countably infinite if it is countable and infinite. This is of course, just another way of saying that A ≈ N. Theorem 1.7 Suppose A 6= ;. The following are equivalent: 1. A is countable 2. card(A) ≤ @0 3. There is a surjection f from N onto A Theorem 1.8 If A and B are countable then (1) A × B and (2) A [ B are countable Cor 1.3 If A1,. ,An are countable then A1 × A2 × · · · × An and A0 [···[ An are countable Theorem 1.9 A countable union of countable sets is countable. 3 How do we prove properties like these? Start with: An alternate approach to countability: Definition 1.5 If Σ is any set (which we think of as a set of \letters", let Σ∗ be the set of finite \words" on Σ; formally, Σ∗ := S nΣ n2N Theorem 1.10 If Σ is countable then so is Σ∗ Proof: Without loss of generality Σ ⊆ ! − f0g. Define φ :Σ∗ ! ! by φ(τ) = τ(0) τ(1) τ(n−1) p0 p1 ··· pn−1 , where n = domain(τ) and p0; p1;::: enumerates the prime numbers. It is easy to see that φ is one-to-one, and that suffices. Cor 1.4 Q is countable Proof: Every rational can be written as a word on the finite alphabet −37 f0; 1; 2; 3; 4; 5; 6; 7; 8; 9; =; −}; eg, 4 is the 5 character word \-37/4". Cor 1.5 if A and B are countable then A × B is countable Proof: Every pair in A × B can be writen as a word on the alphabet A [ B [ f(; );; g (note the comma in the last set). 2 Well-orderings and Transfinite Induction Recall that a set X with a binary relation ≺ is a (strict) linear order provided (X; ≺) satisfies: Transitivity For all x; y; z 2 X, if x ≺ y and y ≺ z then x ≺ z Trichotomy For eny x; y 2 X exactly one of the following hold: x ≺ y; x = y; y ≺ x Examples: 1. N; Z; Q; R under the usual ordering 2. R2 (or C) is not linearly ordered under the ordering (x; y) < (r; s) provided x < r and y < s 3. R2 (or C) is linearly ordered under the ordering (x; y) < (r; s) provided x < r or (x = r and y < s) 4. N2 under this same order 5. ajb is not a linear ordering on N+ Definition 2.1 A linearly ordered set (X; ≺) is well-ordered provided every nonempty subset of X has a least element. Which of the sets above is well-ordered? Theorem 2.1 (AC) Every set can be well-ordered 4 Well-ordered sets have some fantastic properties: 1. Every WO set (X; ≺) has a least element. 2. In fact, an infinite WO set starts with a copy of N. 3. If x is not the largest element of X then x has an immediate successor. 4. A WO set cannot be embedded onto a proper initial segment of itself. Definition 2.2 If (X; ≺) is a WO set write x0 =least element of X. If x; y 2 X then (x; y) = fz 2 X : x ≺ z ≺ yg Similar definitions for [x; y); [x; y]; etc. An initial segment of (X; ≺) is an interval of the form [x0; x) or [x0; x]. An embedding is a 1-1 order-preserving function. More nice properties: 1. If (X; ≺) and (Y; C) are WO sets, then either X embeds onto an initial segment of Y or vice-versa. (So any two WO sets are comparable.) 2. If we identify any two WO sets that can be put into an order-preserving 1- 1 correspondence (\order type"), then the collection of all WO sets can be ordered by \X < Y if X embeds onto a proper initial segment of Y ", and the class of all such WO sets is itself linearly ordered, in fact well-ordered! 2.1 Transfinite Induction Theorem 2.2 Let (X; ≺) be a WO set with least element x0. 1. Suppose S ⊆ X satisfies two properties: (a) x0 2 S; (b) For any x 2 X, if [x0; x) ⊆ S then x 2 S. Then S = X 2. Suppose P (x) is a property of elements of X and P satisfies two properties: (a) P (x0); (b) For any x 2 X, if (8y ≺ x)P (y) then P (x). Then (8x 2 X)P (x) Example: A WO set (X; ≺) can be order-embedded into R (with the usual ordering) if and only if X is countable. 5 2.2 Transfinite Recursion We can also define objects like functions using a transfinite form of recursion. Recall how we can define sequences (functions on N) by recursion: Fib. numbers: a0 = 1; a1 = 1; an = an−1 + an−2 for n > 1 2 Sum of squares a0 = 0; an = an−1 + n In both these cases we have some starting value(s), and then a procedure for defining f(n) from the previous values f(0); f(1); ··· ; f(n − 1) together with n. For transfinite recursion we begin with a WO set (X; ≺), and a procedure for defining f(x) from f [x0;x) together with x, and this gives a function defined on X. Examples: Ordinals: Given (X; ≺) define E(x) = fE(y): y < xg for x 2 X. (Note that this implicitly defines E(x0) as well.) The range of E is called an ordinal number.