Lecture 4 Infinite Cardinals Some Philosophy What is

Lecture 4 Infinite Cardinals

Some Philosophy: What is “ 2”? n Definition 1: 2 = 1+1. This actually needs the definition of “ 1” and the definition of the “+” operation. n 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. n Definition 3: 2 = The class of all sets of size 2 (this is indeed a very large class) n Definition 4: 2 = {0, 1}, where 1 = {0} and 0 = {}. Note: 2 is a particular set of size 2.

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

Some Definitions n The size of a set A is less than or equal to that of B, written A B iff there is an injective (oneto-one) function f: A B. n 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. n We say that the set A is equipotent (or equinumerous) with B. n 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 n Obviously: A B. n The relation on sets is: Reflexive, i. e. for all sets A, A A. n Symmetric, i. e. for all sets A and B; A B B A. n Transitive, i. e. for all sets A, B and C; (A B and B C) A C. n The above properties are easy to prove. n Thus, is an equivalence relation on the class of all sets. n

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

Cantor–Bernstein–Schroeder Theorem: n For all sets A and B; (A B and B A) A B. n Proof Outline: We have two injective functions f : A B and g : B A. We use these to construct a bijection h : A B. n The idea is to find a suitable subset C A, such that n (see http: //en. wikipedia. org/wiki/Cantor%E 2%80%93 Bernstein%E 2%80%93 Schroeder_theorem )

The following infinite sets are countably infinite (i. e. equipotent with N): n The set of even numbers n The set of prime numbers (or in general any infinite subset of N) n The set of rational numbers n The set of algebraic numbers (see n The set of computable reals (see n The set computer programs n The set of computer files http: //en. wikipedia. org/wiki/Algebraic_numbers http: //en. wikipedia. org/wiki/Computable_number ) )

Is every infinite set equipotent with N? n Answer: No! the set of reals R is larger than N. n Proof: Clearly 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, …. n We can then construct a real number y distinct from any infinite list of real numbers by letting: the ith digit of y the ith digit of xi

Picture Change all digits of the diagonal. 4 8 2 0 8 2 1 9 0 …. 7 0 7 3 6 9 6 3 9 …. 1 9 6 3 2 9 4 9 2 …. 0 7 4 9 3 8 9 5 1 …. 9 3 2 8 9 4 2 9 0 …. 4 6 8 5 3 2 8 0 0 …. 3 6 8 0 5 6 2 1 8 …. 7 5 3 7 8 0 8 1 3 …. 0 8 7 4 2 8 6 8 0 …. . . … … … … …

Picture to get the number: . 517003321…. 5 8 2 0 8 2 1 9 0 …. 7 1 7 3 6 9 6 3 9 …. 1 9 7 3 2 9 4 9 2 …. 0 7 4 0 3 8 9 5 1 …. 9 3 2 8 0 4 2 9 0 …. 4 6 8 5 3 3 8 0 0 …. 3 6 8 0 5 6 3 1 8 …. 7 5 3 7 8 0 8 2 3 …. 0 8 7 4 2 8 6 8 1 …. . . … … … … …

In general: Cantor’s Theorem n For every set A, its power set defined by n n 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) n Answer: Our Mathematics is too weak to decide this question (assuming it’s consistent)! n n n

Cardinal Arithmetic n n n n Definition: Let |A| = , and |B| = . We define: + = |A B| (if they are disjoint) = |A B| = |BA|, where BA 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