Math 455 Some notes on Cardinality and Transfinite Induction
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1 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, A n, A B =the set of all functions from A into B Identify a natural number with the set of its predecessors, n = {0, 1,..., n 1}. Today: 0 = {} 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: N Z N 2 Q >0 Q R P(N) R 2 {f : R R : f is continuous} 1
2 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) 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 = {x A : x / g(x)} Claim: B / range(g). To see this, let x 0 A, and consider 2 cases: (i) x 0 g(x 0 ); then x 0 / B, so B g(x 0 ). (ii) x 0 / g(x 0 ); then x 0 B, so B g(x 0 ). Either way, B g(x 0 ). Since x 0 was arbitrary in A, B / range(g). Since g was arbitrary, A P (A). (Note that a {a} 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) A 2 = the set of functions from A into {0, 1} Proof: Identify every subset of A with its characteristic function. Cor 1.1 N R 2
3 1.1 Finite sets Definition 1.3 A is finite if m 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 B A A B, Dedekind-finite otherwise Theorem 1.6 Suppose A. 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 N 5. There exists a surjection g : m A for some m N Cor 1.2 If A is finite then there is a unique m 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) = ℵ 0, and we write card(a) ℵ 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. The following are equivalent: 1. A is countable 2. card(a) ℵ 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 A 1,...,A n are countable then A 1 A 2 A n and A 0 A n are countable Theorem 1.9 A countable union of countable sets is countable. 3
4 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 := n Σ Theorem 1.10 If Σ is countable then so is Σ Proof: Without loss of generality Σ ω {0}. Define φ : Σ ω by φ(τ) = p τ(0) 0 p τ(1) 1 p τ(n 1) n 1, where n = domain(τ) and p 0, p 1,... 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 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, /, }; eg, 37 4 is the 5 character word -37/4. n N 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 {(, ),, } (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 X, if x y and y z then x z Trichotomy For eny x, y X exactly one of the following hold: Examples: 1. N, Z, Q, R under the usual ordering x y, x = y, y x 2. R 2 (or C) is not linearly ordered under the ordering (x, y) < (r, s) provided x < r and y < s 3. R 2 (or C) is linearly ordered under the ordering (x, y) < (r, s) provided x < r or (x = r and y < s) 4. N 2 under this same order 5. a b 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
5 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 x 0 =least element of X. If x, y X then (x, y) = {z X : x z y} Similar definitions for [x, y), [x, y], etc. An initial segment of (X, ) is an interval of the form [x 0, x) or [x 0, x]. An embedding is a 1-1 order-preserving function. More nice properties: 1. If (X, ) and (Y, ) 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 x Suppose S X satisfies two properties: (a) x 0 S; (b) For any x X, if [x 0, x) S then x S. Then S = X 2. Suppose P (x) is a property of elements of X and P satisfies two properties: (a) P (x 0 ); (b) For any x X, if ( y x)p (y) then P (x). Then ( x 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
6 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: a 0 = 1; a 1 = 1; a n = a n 1 + a n 2 for n > 1 Sum of squares a 0 = 0; a n = a n 1 + n 2 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) = {E(y) : y < x} for x X. (Note that this implicitly defines E(x 0 ) as well.) The range of E is called an ordinal number. Vector spaces: Let V be a vector space (an abstract one, not just R n ). Recall that a basis for V is a set B V such that B is a linearly independent (LI) set (no nontrivial linear combination of finitely many elements of B is the zero element 0) whose linear span is all of V, V = span(b) =the set of finite linear combinations of elements of B. The every vector space has a basis. Proof by TF recursion: 6
7 3 Some Equivalents of the Axiom of Choice 1. If A is a set of disjoint nonempty sets then there is a set B such that for every a A B a is a singleton. 2. If R is a relation then there is a function F R with dom(r) = dom(f ) 3. If {A i } i I is an indexed collection of nonempty sets, then there exists a function φ with domain I such that φ(i) A i for all i. 4. If {A i } i I is an indexed collection of nonempty sets, then Π i A i is nonempty. 5. (Cardinal Comparability) For every C, D there is either an injection from C into D or an injection from D into C. (In other words, either card(c) card(d) or card(d) card(c).) 6. (Zorn s Lemma) Let A be a set of sets, suppose A is closed under unions of chains. Then A contains a maximal element. 7. Every set can be well-ordered. 7
8 4 Exercises: 1. Prove that the set of polynomials in x with rational coefficients is countable. 2. Prove that the set of all matrices with rational entries is countable. 3. A real number is computable if there exists a computer program that will list its digits. (That is, you run the program, it starts to print out something like , and if you never stop the program the final number is said to be computable.) Show that the set of all computable real numbers is countable. 4. Let A be a collection of disks in the plane, no two of which intersect. Show that A is countable. (A disk is a circle together with its interior. What if A consists of circles, not disks?) 5. Show that any well-ordered subset of R is countable. 6. (Research problem.) Look up Hamel basis, and see how it is used to construct an example of a function on R satisfying ( x, y R) f(x + y) = f(x) + f(y) but which is not continuous anywhere. 7. (Research problem.) Take any result from a 400-level Algebra or Topology course usually proved using Zorn s Lemma or the Hausdorff Maximality Property, and prove it instead using transfinite induction. 8
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