Great Theoretical Ideas In Computer Science V. Adamchik CS 15-251 Outline Lecture 20 Carnegie Mellon University Cantor’s Legacy Cardinality Diagonalization Continuum Hypothesis Cantor’s theorem Cantor’s set Cantor (1845–1918) Galileo (1564–1642) Galileo: Dialogue on Two New Sciences, 1638 If I should ask further how many squares there are one might reply truly that there are as many as the corresponding number of square- Salviati Simplicio roots, since every square has its own square-root and every square-root its own square… I take it for granted that you know which of the numbers are squares … Neither is the number of squares less I am quite aware that a and which are not. than the totality of all the numbers, … squared number is one which results from the … nor the latter greater than the former, … multiplication of another number by itself. Very well… If I assert that all numbers, … and finally, the attributes “equal,” including both squares and non-squares, “greater,” and “less,” are not applicable are more than the squares alone, I shall to infinite, but only to finite, quantities. speak the truth, shall I not? Let’s review this argument Cantor’s Definition ℕ = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … } Sets A and B have the same S = { 0, 1, , , 4, , , , , 9, … } ‘cardinality’ (size), written |A| = |B|, if there exists a bijection between them. “All numbers include both S ⊊ ℕ squares and non-squares.” “Every square has its own There is a square-root and every bijection square-root its own square…” between ℕ and S. 1 Reminder: what’s a bijection? Cantor’s Definition It’s a perfect matching between A and B. It’s a mapping f : A → B which is: Sets A and B have the same an injection ‘cardinality’ (size), written |A| = |B|, (i.e., ‘one-to-one’: f(a)≠f(b) if a≠b) if there exists a bijection between them. & a surjection (i.e., ‘onto’: ∀b∈B, ∃a∈A s.t. f(a)=b). E.g.: |ℕ| = |Squares| It’s a function f : A → B which has an because the function f : ℕ → Squares inverse function, f−1 : B → A (also a bijection). defined by f(a)=a2 is a bijection. If A and B are infinite sets Do N and E have the same cardinality? do we always have |A| = |B|? N = { 0, 1, 2, 3, 4, 5, 6, 7, … } E = { 0, 2, 4, 6, 8, 10, 12, … } That’s exactly what I was wondering in 1873… f(x) = 2x is 1-1 onto. Let’s try some examples! Do N and Z have the same cardinality? Transitivity Lemma N = { 0, 1, 2, 3, 4, 5, 6, 7, … } Lemma: If f: AB is 1-1 onto, and Z = { …, -2, -1, 0, 1, 2, 3, … } g: BC is 1-1 onto. Then h(x) = g(f(x)) defines a function h: AC that is 1-1 onto f(x) = x/2 if x is odd -x/2 if x is even Hence, N, E, and Z all have the same cardinality. The odd numbers in N map to the positive integers in Z. The even numbers in N map to negative integers in Z. 2 A Natural Intuition A Natural Intuition Let’s re-examine the set Z. Consider listing Intuitively, what does it mean to find a bijection them in the following order. between a set A and N ? First, list the positive integers (and 0): Then, list the negative integers: It means to list the elements of A in some order so that if you read down the list, every element will 0, 1, 2, 3, 4, . ., -1, -2, -3, -4, . get read. What is wrong with this counting? If you start reading down the list, you will never actually get to the negatives! A Natural Intuition Do N and ℚ have the same cardinality? How about this list? 0, 1, -1, 2, -2, 3, -3, 4, -4, . N = { 0, 1, 2, 3, 4, 5, 6, 7, …. } ℚ For any integer n, at most |2n| integers come = The Rational Numbers before it in the list, so it will definitely get read. Don’t jump to conclusions! No way! The rationals are dense: between any two there is a third. You can’t There is a clever way to list the rationals, list them one by one without leaving one at a time, without missing a single one! out an infinite number of them. First, let’s warm up with another interesting example: N can be paired with N × N 3 Theorem: N and N × N have the Theorem: N and N × N have the … same cardinality … same cardinality 4 4 6 3 3 The point (x,y) The point (x,y) represents the 7 represents the 2 2 3 ordered pair ordered pair (x,y) 4 (x,y) 1 1 1 8 0 5 0 0 2 9 0 1 2 3 4 … 0 1 2 3 4 … Onto the Rationals! The point at x,y represents x/y The point at x,y represents x/y Cantor’s 1877 letter to Dedekind: “I see it, but I don't believe it! ” N and ℚ have the same cardinality. The rational numbers ℚ are countable. 4 Is {0,1}* countable? Let’s do one more example. Let {0,1}* denote the set of all binary strings of any finite Yes, this is easy. Here is my listing: length. ϵ, 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, Is {0,1}* countable? 110, 111, 0000,… Length 0 Length 1 strings Length 2 strings Length 3 strings strings in binary order in binary order in binary order Thus: The set of all possible Java/C/Py programs is countable. Onto the Reals! The set of all possible finite length pieces of English text is countable. Is ℝ countable? I asked Dedekind this in Cantor proved ℝ is uncountable November, 1873. in December 1873. ℝ = { ½, 2, ,…} To do this, he invented a very important technique called “…if you could prove that ℝ is uncountable, it “Diagonalization” would have an application in number theory: a new proof of Liouville’s theorem that there are transcendental numbers!” 5 Theorem: |R| |N| || Theorem: {0,1}∞ is NOT countable. Suppose for the sake of contradiction that you We will instead prove that can make a list of all the infinite binary strings. the set of all infinite binary strings, ∞ denoted {0,1} , is uncountable. 0: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0… ∞ Suppose {0,1} , is countable, thus there is a 1: 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1… bijection f: N -> R 2: 1 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1… We will show that this is not a surjection. 3: 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0… 4: 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1… … … Theorem: {0,1}∞ is NOT countable. Theorem: {0,1}∞ is NOT countable. Consider the string formed by the ‘diagonal’: the k-th bit in the (k-1)-st string Next, negate each bit of the string on the diagonal: 0: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0… 1 0 0 0 1 0… 1: 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1… 2: 1 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 1… It can’t be anywhere on the list, since it differs 3: 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0… from every string on the list! 4: 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1… 5: 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0… Contradiction. … … ℝ is uncountable. Even the set [0,1] of all reals Continuum Hypothesis between 0 and 1 is uncountable. The cardinality of natural numbers |ℕ| = ℵ0 (aleph zero or naught) This is because there is a bijection There are no infinite sets with cardinality less ℵ0 ∞ between [0,1] and {0,1} . Question: Is there a set S with |ℕ| < S < |ℝ|? ℵ1 is the It’s just the function f which maps each or smallest set larger than ℵ0 ℵ0 real number between 0 and 1 to its ℵ0 < S < 2 = ℵ1? binary expansion! This is called the Continuum Hypothesis. Cantor spent a really long time trying to prove |ℝ| = ℵ1, with no success. 6 The Continuum Hypothesis We know there are at least cannot be proved or disproved 2 infinities. from the standard axioms of set theory! (the number of naturals, the number of reals.) This has been proved! Are there more? Kurt Godel, 1940 Definition: Power Set Cantor's Theorem: The power set of a set has a greater cardinal number than the set itself, S The power set of S is the set of all subsets of S.