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Home , Transfinite induction

Math 455 Some notes on and Transfinite Induction

(David Ross, UH-Manoa Dept. of Mathematics)

1 Cardinality

Recall the following notions: , relation, one-to-one, onto, on-to-one correspondence, injection, surjection, , equivalence relation, An, AB =the 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:

2 >0 N ≈ Z ≈ N ≈ Q ≈ Q

2 R ≈ P(N) ≈ R ≈ {f : R → R : f is continuous}

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 = {x ∈ A : x∈ / g(x)} Claim: B/∈ range(g). To see this, let x0 ∈ A, and consider 2 cases: (i) x0 ∈ g(x0); then x0 ∈/ B, so B 6= g(x0). (ii) x0 ∈/ g(x0); then x0 ∈ B, so B 6= g(x0). Either way, B 6= g(x0). Since x0 was arbitrary in A, B/∈ range(g). Since g was arbitrary, A 6≈ P (A). (Note that a 7→ {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 of Choice (AC).

Proposition 1.1 For every set A, P (A) ≈ A2 = the set of functions from A into {0, 1}

Proof: Identify every of A with its characteristic function.

Cor 1.1 N 6≈ R

2 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 ( 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 ∈ 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) = ℵ0, and we write card(A) ≤ ℵ0 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) ≤ ℵ0

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 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 n∈N 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) = 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 {0, 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 ∪ {(, ),, } (note the comma in the last set).

2 Well-orderings and Transfinite Induction

Recall that a set X with a ≺ 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: 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. 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 . 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 ∈ 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 [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 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 ∈ S;

(b) For any x ∈ X, if [x0, 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 (x0); (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 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) = {E(y): y < x}

for x ∈ X. (Note that this implicitly defines E(x0) as well.) The range of E is called an .

Vector spaces: Let V be a (an abstract one, not just Rn). Recall that a 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 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 3 Some Equivalents of the

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 . 2. If R is a relation then there is a function F ⊆ R with dom(R) = dom(F )

3. If {Ai}i∈I is an indexed collection of nonempty sets, then there exists a function φ with domain I such that φ(i) ∈ Ai for all i.

4. If {Ai}i∈I is an indexed collection of nonempty sets, then ΠiAi 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 4 Exercises:

1. Prove that the set of polynomials in x with rational coefficients is count- able. 2. Prove that the set of all matrices with rational entries is countable. 3. A 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 13.43256. . . , 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.

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