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Appendix A Infinite Sets: Part I: What Does Infinity Mean? So what does infinite really mean? We use the word regularly, but how do we define it? Generally, people would define infinite as "endless", "without beginning or end", “going on forever” (which begs the question, “what does forever mean?”) or “going on and on”. Another response is the somewhat self-referencing, “anything that is not finite”. None of these definitions are rigorous, and in mathematics we usually want rigor. First, we start by recognizing that infinite is an adjective, it is a characteristic that something may have; thus rigorous definitions of infinite in mathematics generally depend on what the adjective infinite is modifying. If you take a Calculus or analysis course you will, most probably, define infinite limits. In this appendix we will define infinite sets. There are two ways to go about defining infinite sets, one easier for most people to follow than the other. We will begin with that one. This approach is to first rigorously define what we mean by finite sets then define infinite sets as sets that are not finite. Let us start with our intuitive idea of a finite set. If a set is finite, then, given enough time, theoretically, we could count its elements. (I say theoretically because, while I believe that the set of grains of sand on the planet is finite, I do not believe it is humanly possible to count them.) So what do we mean when we say, for example, this set has 10 elements? What exactly are we doing when we count? Imagine for a moment that you are a member of an ancient civilization, one that has not yet developed a counting system. Further imagine that you are a shepherd with a collection of sheep and you want to take them out in the countryside to graze then bring them all back home. How would you be sure that they all came back with you, that you didn’t lose any along the way? If your collection of sheep is small enough, you can get to know each sheep personally, give them each a name. Then you might notice that little Sally Ewe must have wandered off because you don’t see her anywhere. But then you need to remember all the names. You might try the buddy system we use with children but sheep don’t usually cooperate or communicate well enough for that. Also, if you were missing an entire pair, how would you know? Basically, you need a way to count them, but without using numbers. One way to do so, if you have the natural resources available, would be to take a large collection of stones and make the sheep pass through a gate one at a time. As each sheep passes through the gate you would place a stone in a box or bowl you have constructed for this purpose. (Ok, so I did use the number one but there is evidence that early civilizations had words for one, two and many before they had developed number systems.) When you return with your sheep, you can again have them pass through the gate taking a stone out of your box or bowl for each sheep. If, when they have all passed through the gate, you have any stones left in your box/bowl, then you have lost a sheep for each remaining stone. Otherwise, you’ve brought them all back. For this discussion we will ignore questions about how you convince the sheep to pass through the gate in an orderly fashion, I'm sure you could come up with something along these lines that would work; Page 288 Discrete Mathematics: A Brief Introduction Appendix A after all you wouldn't be reading a discrete mathematics text if you weren't a problem solver. Now suppose you realize one day that the stone business is rather tedious and besides which, if you want to take your sheep further afield and check that you have them along the way, the stones are rather heavy to carry around. So you decide to invent some special words, which you memorize to represent the stones. You always use the words in exactly the same order so you know which words you have used, just by remembering the last one. Instead of putting the sheep in one-to-one correspondence with stones, you put them in one-to-one correspondence with the words. Now you’re cooking, I mean, counting. Let us now return to the present day where we have a well defined set of “counting numbers”, {1, 2, 3, . . . } and we can make the following definitions: Definition A.1 A one-to-one correspondence between the elements of two sets is a matching where each element of one set is matched with exactly one element of the other where no elements of either set are left out. (note: if you have read section 1.3, you could write a cleaner definition using one-to-one and onto functions) Definition A.2 A set S whose elements can be put in one to one correspondence with the elements of the set {1, 2, . . . ., n} of counting numbers is said to have n elements or to have cardinality n. We write |S| = n. Definition A.3 Two sets A and B are the same size or have the same cardinality if there exists a one-to-one correspondence between their elements. Now we have a relatively straight forward definition of finite: Definition A.4 A set is finite if it has n elements for some counting number n. Now it is perfectly rigorous to say: Definition A.5 A set S is infinite if it is not finite. So are all infinite sets the same size? First, consider the set N of all the counting numbers. Is it infinite? How do you prove it? Theorem A.1: The set N of counting numbers is infinite. We will prove this by contradiction. See section 2.0 and 2.0 problem 3 for a discussion of this method of proof. Before we begin, however, let us note that if there is a correspondence between the elements of {1, 2, . . . ., n} and the elements of a set of distinct numbers K, it is Page 289 Discrete Mathematics: A Brief Introduction Appendix A possible to find the largest element of k using the following algorithm: • • • • • • Algorithm for finding the greatest element in K: Let L be the number corresponding to 1. Let I=1 (*)If the number corresponding to I+1 is greater than L then let L be that number. Let I=I+1 If I # n then go to (*). When the algorithm terminates, L is the greatest element of K. ” Now we can prove theorem A.1: Proof: Suppose N is finite. Then there is a counting number n such that there is a correspondence between the elements of {1, 2, 3, . . ., n} and N. Then there is a greatest element L of N which can be found using the above algorithm. Since L is in N, L is a counting number. So L+1 is a counting number greater than L. This is a contradiction to the fact that L is the greatest element in N. Therefore our initial supposition must be false and the theorem is proved by contradiction. ” Of course, you may say that it is obvious that N is infinite. Generally things that are obvious are the most difficult to prove. Sometimes being completely rigorous in mathematics is a little like dealing with a two-year-old in the “why” stage. You want to say “proof: because I said so!” but you have to bite your tongue. Now let us consider the set of even counting numbers, E = {2, 4, 6, . . .}; is it the same size as N? Instinct may have you saying “no” since every element of E is in N but not the other way around. Clearly this says there are more elements in N? We have an obvious correspondence (not one-to-one) which pairs everything in E with something in N leaving nothing in E out but leaving infinitely many elements of N out. But beware! Just because one correspondence doesn’t work doesn’t mean there isn’t one that does. Consider the following mapping: 1->2 2->4 3->6 4->8 ! n->2n ! matching every element of N with its double in E. This is a one-to-one correspondence between Page 290 Discrete Mathematics: A Brief Introduction Appendix A N and E. (Verification is left as an exercise.) So E and N are the same size, even though E is a proper subset of N! (For a discussion of proper subsets see section 1.0.) You can’t do that with finite sets! (try) which leads to an alternate definition for infinite: Definition A.4(alternate) A set S is infinite if it can be put in one-to-one correspondence with a proper subset of itself. And for finite: Definition A.2 (alternate) A set S is finite if it is not infinite. It is left as a challenge to the reader to prove1 that the two pairs of definitions (our originals and the alternates) are equivalent. Now let’s return to our earlier question, “are all infinite sets the same size?” A way to rephrase this would be: “can all infinite sets be put in one-to-one correspondence with the set of counting numbers?” Well, any set whose elements can be listed in order, say from largest to smallest or any other linear order could since you could correspond the first element in the list with 1, the second with 2, etc. So, if there is an infinite set that cannot be put in one-to-one correspondence with the counting numbers then it must be one that cannot be put in order from largest to smallest. The integers don’t have a smallest element but they can be listed if you start at 0 and alternate between positive and negative: 0, -1, 1, -2, 2, -3, 3, . . .. Let’s consider the set of all positive fractions of counting numbers; they can’t be put in order from smallest to largest either, so you might be tempted to conclude that there are more of them than there are counting numbers. But remember, just because one method of listing them won’t work, don’t assume none will. Consider the following doubly infinite array of fractions: 1/1 2/1 3/1 4/1 5/1 6/1 ... 1/2 2/2 3/2 4/2 5/2 6/2 ... 1/3 2/3 3/3 4/3 5/3 6/3 ... 1/4 2/4 3/4 4/4 5/4 6/4 ... 1/5 2/5 3/5 4/5 5/5 6/5 ... 1/6 2/6 3/6 4/6 5/6 6/6 ... ! ! ! ! ! ! " 1 A proof of this requires choosing an element from the infinite set. There is actually an axiom in mathematics called the Axiom of Choice which allows us to do this. If one does not accept the axiom of choice then this equivalence would not necessarily hold. Page 291 Discrete Mathematics: A Brief Introduction Appendix A Each positive rational number will appear in the array (in fact it will appear infinitely many times). Now, notice (see figure 1) that you can draw a path through the array, which, if continued, will visit each rational number, one at a time. When you reach an entry that is not in reduced form (circled below), simply skip it and pass to the next. The numbers you visit, in order along this path provide list of the rational numbers and a blueprint for putting them in one to one correspondence with the counting numbers. 1/1 2/1 3/1 4/1 5/1 6/1 .. . 1/2 2/2 3/2 4/2 5/2 6/2 .. . 1/3 2/3 3/3 4/3 5/3 6/3 .. . 1/4 2/4 3/4 4/4 5/4 6/4 .. . 1/5 2/5 3/5 4/5 5/5 6/5 .. . 1/6 2/6 3/6 4/6 5/6 6/6 .. . ! ! ! ! ! ! " The list this provides begins: 1/1, 2/1, 1/2, 1/3, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 5/1, . . .and will contain all positive rational numbers exactly once. For ease of discussion, let us define a term for sets that can be put in one-to-one correspondence with the counting numbers: Definition A.6: Any set that can be put in one-to-one correspondence with a subset of the natural numbers is said to be countable. Infinite sets that can be put in one-to-one correspondence with the set of natural numbers are said to be countably infinite or denumerable. So all finite sets are countable; the integers are countable; and the rational numbers are countable. Are there any sets that are not countable? This discussion would not be very interesting if the answer were no, so you have probably guessed that there are. Consider the set of real numbers in the interval [0,1] that is, real numbers x such that 0#x#1. At the end of the 19th century a controversial mathematician named Georg Cantor (1845-1918) developed a diagonal process (the Cantor Diagonal Process) to show that this set is not denumerable. Page 292 Discrete Mathematics: A Brief Introduction Appendix A Cantor's was interested in medieval theology2, particularly its arguments involving the continuous and the infinite, this led him to develop a theory of infinite sets which resulted in some disagreement in mathematical philosophy. His biggest critic was Leopold Kronecker after whom the Kronecker delta "function", another controversial tool used extensively by physicists, was named. In any case, Cantor's diagonal process provides a now widely accepted method of proving the following: Theorem A.2: The set of real numbers between 0 and 1 is not countable. Proof (using the Cantor Diagonal Process): Suppose that this set were denumerable, then there would be an infinite list of real numbers between 0 and 1 which would contain them all. But, we can show that, given any list of these numbers, there is an algorithm for constructing a number not in the list. The algorithm relies on the fact that every real number between 0 and 1 has a decimal expansion, all of whose non-zero digits lie to the right of the decimal point and that every non-zero decimal expansion all of whose non-zero digits lie to the right of the decimal point is a representation for a real number between 0 and 1. Furthermore, if we do not allow expansions ending in an infinite string of nines, this representation is unique. Here is the algorithm: Algorithm for constructing a real number d, not in a given list of real numbers: Let d be the number whose decimal expansion is given by: 0.d1d2d3 . . . where the digits di are defined as follows: If the first digit to the right of the decimal point of the first number in the list is a 1 then d1 = 2, otherwise d1 = 1. If the second digit to the right of the decimal point of the second number in the list is a 1 then d2 = 2, otherwise d2 = 1. And so forth. Thus the digit di is a 1 unless the ith digit of the ith number is a 1 in which case di = 2. Can you see that the number d constructed in this manner is not on the list? This algorithm shows that in any list of real numbers between 0 and 1, there is always at least one number missing. Therefore, there is no complete list of these numbers. Thus there are infinite sets of different sizes and the set of real numbers has a cardinality larger than that of the ” set of integers. We generally use the symbol !0, read Aleph naught, using the first letter in the Hebrew alphabet with the subscript 0, to represent the cardinality of countably infinite sets. The cardinality of the real numbers is sometimes referred to as c for continuum and sometimes referred to as !1. 2 See An Introduction to the History of Mathematics, 6th ed. by Howard Eves; Saunders College Publishing: New York. pg 569. Page 293 Discrete Mathematics: A Brief Introduction Appendix A Writing, discussion, comprehension and exploration (WDCE)exercises: 1. Explain in your own words what infinite means. 2. Explain why the number d constructed in the algorithm used to prove theorem A.2 is not in the list. 3. Research the Axiom of Choice. What is it? Is it or was it ever controversial? Problems: 1. (Challenge) Prove that the two definitions of an infinite set are equivalent (i.e. that each implies the other) point out where the Axiom of Choice comes in. 2. Construct an alternate algorithm for the proof of theorem A.2. 3. Show that if S and T are denumerable sets then ScT is denumerable. (hint: consider the argument that the integers are countable.) 4. State an alternative definition of one-to-one correspondence using one-to-one and onto functions. Part II: Infinitely Many Infinities and the Continuum Hypothesis. (This section of the appendix is best read after reading through Section 2.1.) So we know there are at least two infinities, but are there more? How many? It turns out that there are infinitely many different sizes of infinity. recall, from section 2.1 that if a set has n elements then it has 2n subsets. Definition A.7: (a.k.a.Definition 1.0.8) The set of all subsets of a set A is called the Power Set of A and is denoted P (A). What about the size of the power set of an infinite set, A. Can it be put in one-to-one correspondence with the set itself? It turns out that the answer to this is no. Since there is an obvious one-to-one correspondence between the elements of A and the single-element subsets in P (A), then this means that the power set is bigger than the original set. Theorem A.3: There is no one-to-one correspondence between a set and its power set. The proof of this theorem is reminiscent of the famous paradox of Bertrand Russell (1872-1970) regarding the set of all sets that do not contain themselves. It goes as follows: Page 294 Discrete Mathematics: A Brief Introduction Appendix A Proof: The statement is clearly true for the empty set as it has no elements but its power set has one element. Now suppose there is a one-to one correspondence between a non-empty set A and its power set P (A). Then given an element x there is a subset of A to which x corresponds, we will call this subset p(x). Now make the following definition: if x is in A then x is normal if x0p(x) and otherwise it is not normal. Clearly each element in A is either normal or not normal, thus one of the subsets of A contains all the elements of A that are not normal. Let us denote this subset N. So there must be an element of A, call it n, which corresponds to this set. Furthermore, n is either normal or it is not normal. Which is it? If it is not normal then it is in N, in which case it is normal, which is a contradiction. Therefore, as the original assumption leads to a contradiction, it is not possible to put a set in one-to-one correspondence with its power set. ” Now we know that there are denumerably many infinities, at least. Given the cardinality of the counting numbers, !0, then define !1 to be the cardinality of P (N). !2 to be the cardinality of P (P (N)), etc. (You may have guessed at this point that it is possible to show that the set of real numbers and the set of subsets of the natural numbers are the same size.) In this way we can construct an infinite sequence of increasing cardinalities. These infinite cardinalities are also know as transfinite numbers. And there is a branch of mathematics which studies transfinite numbers. Now we know that there are a countably infinite number of transfinite numbers, but are there more? In particular, are there any sets whose cardinality lies between !0 and !1 or, more generally, between !i and !i+1? The famous Continuum Hypothesis, conjectures not. In about 19403, the Austrian logician, Kurt Gödel, famous for his incompleteness theorem, proved that the continuum hypothesis is consistent with the accepted axioms of set theory, if they are consistent with each other. In other words, he showed that it would be impossible to disprove the continuum hypothesis in the standard realm of modern set theory. He also conjectured that it would be impossible to prove it as well. In 1963, Paul J. Cohen of Stanford University proved he was right. Thus the Continuum Hypothesis is undecidable on the basis of the axioms of set theory. It is to the axioms of set theory, like the parallel postulate is to the rest of the axioms of Geometry. Writing, discussion, comprehension and exploration (WDCE)exercises: 1. Investigate Russell's paradox either in the library or on the internet. What can you find out about it? Explain how it relates to Russell's story about a male barber who shaves all the men and only the men who do not shave themselves. Who shaves the barber? 3 See An Introduction to the History of Mathematics, 6th ed. by Howard Eves; Saunders College Publishing: New York. pg 617. Page 295 Discrete Mathematics: A Brief Introduction Appendix A 2. A consequence of theorem A.3 is that there is no largest infinite set. Cantor, however, was bothered by this consequence; what about the set of all sets? Can you reslove this paradox? Explain. 3. The result that the Continuum Hypothesis is undecidable on the basis of the axioms of set theory may make you a little uncomfortable. You may feel that mathematicians have been remiss in developing a theory with insufficient axioms to answer all its questions. In their defense I offer Gödel's Incompleteness Theorem in which Kurt Gödel proved that no set of axioms could ever be sufficient to determine the answers to all questions. What can you find out about this theorem and people's responses to it? What do you think about this result? 4. Can you find or construct a proof that the set of real numbers and the power set for the natural numbers are the same size? 5. What do you know about the parallel postulate? How is it similar to the Continuum Hypothesis? Page 296