Lecture 4 Infinite Cardinals

Some Philosophy: What is “2”? Definition 1: 2 = 1+1. This actually needs the definition of “1” and the definition of the “+” operation. Definition 2: Start with the concept of “two apples”, and remove all aspects of a single apple, e.g. redness, taste, etc.. You’ll be left with the number “2”. This definition is a bit problematic. Definition 3: 2 = The class of all sets of size 2 (this is indeed a very large class) Definition 4: 2 = {0,1}, where 1 = {0} and 0 = {}. Note: 2 is a particular set of size 2.

Some History: What is “n”? Historically, people could not count beyond some (relatively small) finite number, e.g. 10. A (large) number “n” did not have a name, but people could access it by having a bag with n stones. If a shepherd wants to make sure the number of sheep was n, he matches the sheep with the stones. Thus, two sets have the same size if there is a bijection (one-to-one correspondence) between the elements of the sets..

Some Definitions The size of a set A is less than or equal to that of B, written A  B iff there is an injective (one-to-one) function f:A  B. The size of a set A equals the size of the set B, written A  B iff there is a bijective (one-to-one and onto) function f:A  B. We say that the set A is equipotent (or equi- numerous) with B. Note: If A is finite and has n elements, we can take the size of A = n. However, the size of an infinite set A is yet to be defined.

Some Simple Facts about  Obviously: A  B  A  B. The relation  on sets is: Reflexive, i.e. for all sets A, A  A. Symmetric, i.e. for all sets A and B; A  B  B  A. Transitive, i.e. for all sets A,B and C; (A  B and B  C)  A  C. The above properties are easy to prove. Thus,  is an equivalence relation on the class of all sets.

Some Simple Facts about  The relation  on sets is: Reflexive, i.e. for all sets A, A  A. Transitive, i.e. for all sets A,B and C; (A  B and B  C)  A  C. Antisymmetric, i.e. for all sets A,B; (A  B and B  A)  A  B. The first two properties are easy to prove, the third constitutes an important theorem…

Cantor–Bernstein–Schroeder Theorem: For all sets A and B; (A  B and B  A)  A  B. Proof Outline: We have two injective functions f :A  B and g :B  A. We use these to construct a bijection h :A  B. The idea is to find a suitable subset C  A, such that (see )

The following infinite sets are countably infinite (i.e. equipotent with N): The set of even numbers The set of prime numbers (or in general any infinite subset of N) The set of rational numbers The set of algebraic numbers (see ) The set of computable reals (see ) The set computer programs The set of computer files

Is every infinite set equipotent with N? Answer: No! the set of reals R is larger than N. Proof: Clearly N  R. N  R means that there is a bijection f: N  R, i.e. a listing of all reals of the form x 1,x 2,x 3,…. We can then construct a real number y distinct from any infinite list of real numbers by letting: the i th digit of y  the i th digit of x i

Picture Change all digits of the diagonal … … … … … … … … …...………………………

Picture to get the number: … … … … … … … … … …...………………………

In general: Cantor’s Theorem For every set A, its power set defined by P(A) = {X: X  A} is larger than A. Proof: Clearly A  P(A). If A  P(A), then there is a bijection f: A  P(A). However, the subset B of A defined by: B = {a  A: a  f(a)} is not covered by f. If it were, i.e. B = f(a), for some a, then: a  B  a  f(a)  a  B, a contradiction.

The Continuum Hypothesis We have infinitely many infinities. We call these א 0, א 1, א 2,… (the alephs) These are the infinite cardinal numbers א 0 (called aleph_0) denotes the size of N. We say: |N| = א 0 Question: Is |R| = א 1 ? (This is the so-called Continuum Hypothesis) Answer: Our Mathematics is too weak to decide this question (assuming it’s consistent)!

Cardinal Arithmetic Definition: Let |A| = , and |B| =. We define:  + = |A  B| (if they are disjoint)   = |A  B|  = | B A|, where B A is the set of all functions from B to A. Note: These definition generalize the arithmetic of natural numbers. Facts: If one of  and is infinite, then:  + =   = max{ , } If  , then  = 2 >

Thank you for listening. Wafik