Section 2.6 Cantor s Theorem and the ZFC Axioms


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1 1 Section 2.6 and the ZFC Axioms Purpose of Section: To state and prove Cantor theorem, which guarantees the existence of infinities larger than the cardinality of the continuum. We also state and discuss and ZermeloFrankel axioms and the Axiom of Choice. Introduction Cantor must have felt he was on a great adventure. He had found sets of infinite size, some countable with cardinality ℵ and some uncountable with cardinality c. He then wondered if there were larger sets. That is a larger infinities than the continuum. Cantor s Discovery of Larger Sets Often little ideas lead to big ideas. We have seen that for finite sets the power set of a set is always larger than the set itself. For example, the set A = a, b, c has three elements, whereas the power set has ( ) = φ,,,,{, },{, },{, },{,, } P A a b c a b a c b c a b c = elements. In other words, for finite sets A we have A P ( A) <. This prompted Cantor to ask if the same principle was at work for infinite sets. This question was answered in the affirmative by the following theorem. Theorem 1 () 1 () The power set ( ) strictly larger cardinality than the set A. P A of any set A has a Proof: As for many of Cantor s theorems, the proof proceeds by contradiction. The assumption that there exists a onetoone correspondence between A and P ( A ) leads to a contradiction. We first prove the result when A is countable, which allows one to get a good visual grasp of the proof. We then prove the result uncountable sets. If A = N = A is countable, then without any loss of generality we let 1,2,3,..., our favorite infinite countable set, and assume there exists a onetoone correspondence N P( N ). To get a good feel for the proof, we list a few elements of P ( N ) : ( N ) P =, 3, 6, 3,5, 7, 5,9,13, 9,11,15,...,...
2 2 and attempt to pair off elements of N with elements of P ( N ) as shown in Table 1: N c( N ) 1 { 3,5 } 2 { 1,2,7 } 3 { 9,13,2 } 4 { 1,5,6 } 5 { 2,5,11, 23 } Trial Pairing of the Natural Numbers and Its Power Set Table 1 Note that the numbers 1,3,4 are not members of the subset which they are paired with, while 2 and 5 do belong to the sets in which they are paired. We call the numbers 1,3,4 uncaptured numbers, and the numbers 2 and 5 captured numbers. We now create a rogue subset of the natural numbers (a member of P ( N ) ) that is not matched with any natural number, thus contradicting the fact there is a onetoone correspondence between N and P N. This rogue set is the set of all unmatched numbers. its power set ( ) That is { : does not belong to the set it is matched to} U = n N n In our correspondences in Table 1, we have U = { 1,3,4,... }. But U is a subset of natural numbers so it must be paired with some natural number, say, let s say n U as we have illustrated in Table 2. N c( N ) 1 { 3,5 } 2 { 1,2,7 } 3 { 9,13,2 } U = 1,3,4,..., n,... n n U or n U? Table 2
3 3 =? If you say n U We now ask the question? Is n U { uncaptured numbers} that means n is an uncaptured number, but n U { 1,3, 4,..., n,... } = that means n is captured or n U, which contradicts n U. On the other hand if you say n U n U = 1,3, 4,... which that means n is an captured number, but means n is uncaptured or n U, which contradicts n U. In either case a P N P N. But we can t contradiction. Hence, we conclude that N ( N ). ( ) have P ( N) < N since the power set P ( N ) contains the singletons m of members m N, and these singletons form a copy inside P ( N ) of elements of N, and so the only remaining possibility is N < P ( N ), which proves Cantor s theorem for countable sets. Uncountable Sets: The proof for uncountable sets is the basically same as for countable sets; we just don t have the nice pictures to help with our understanding. Let f be any function from a set A to its power set P ( A ). We show that f is not onto the power set by exhibiting a subset of A not in the range of f. The subset we find is our rogue set discussed earlier: { : ( )} ( ) U = x A x f x P A. To show U is not in the range of f, suppose U is in the range of f. This means that for some y A, we have f ( y) U. Now ask whether y U or y U. If y U then y f ( y), but that implies, by definition of U, that y U. On the other hand if y U, then y f ( y) and therefore y U. In both cases whether y U or y U we arrive at a contradiction. Hence, we cannot assume that f maps A onto its power set. Hence, Cantor s theorem is proven for an arbitrary infinite sets 1. Margin Note: Cantor presented the proof of what we now call Cantor s Theorem in 1891 Űber eine elemenare Frage der Mannigfaltigkeitslehre, which was the first appearance of Cantor s Diagonal argument, used in the proof that the real numbers are uncountable. Summary: Since the power set of a set, finite or infinite, always has more members than the set itself, it is always possible to find a set with larger cardinality by simply taking the power set of a given set. Thus, one can obtain a sequence of sets having larger and larger infinities ad infinitum 1 We set is interpreted using the ZermeloFrankel axioms of set theory. We have seen that logical paradoxes arise when sets are chosen without regard to any conditions.
4 4. ( ) ( ) ( ) ( ) ( ( )) ℵ = N < P N < P P N < P P P N < all uncountable except for smallest infinity ℵ. Cantor called these infinite numbers transfinite numbers, which means numbers larger than the finite numbers. Since Cantor s theorem implies that for every set there is a greater set, from which it follows there is not set of all sets. That is, there is no set of everything 2. Now that we know the power set of the natural numbers is larger than the natural numbers itself, just how big is it? The following theorem answers that question. Theorem 2 ( c P( ) = N ) The cardinality of the power set of natural numbers is equal to the cardinality of the real numbers. That is c P( ) = N. Proof Since the real numbers and the open interval (,1 ) both have the same cardinality ( c ), we find a onetoone correspondence P ( N ) (,1). We first make the observation that any real number x (,1) can be expressed in binary decimal form x =. b b b where each b j is or 1. For example x =.111 = = We now associate to each real number x =. b1b 2 the subset of natural numbers consisting of the indices j for which b j = 1. For example, we associate. x =.111 1,3, 4,... x = ,3,5,6,... x = ,2,3,5,... which gives a onetoone correspondence (,1) P ( ) N. We denote 2 ℵ to stand for the cardinality of the power set of the natural numbers, which from Example 1 we have seen is the cardinality of the continuum, i.e. ℵ c = 2. 2 The great German mathematician David Hilbert said in 191 that "No one shall drive us from the paradise which Cantor created. Later in 1926 Hilbert is quoted as saying "It appears to me to be the most admirable flower of the mathematical intellect and one of the highest achievements of purely rational human activity."
5 5 If we now call ℵ 1 the next larger infinite number after ℵ, an interesting question arises. We know ℵ is the smallest infinity and that the numbers have a larger cardinality c, but are we sure there isn t an infinite number between the two. In other words, is there some mystery set (or sets) M that have cardinality larger than ℵ but less than c. In other words ℵ < M < c. Stated another way, is c = ℵ 1? Cantor believed the assertion was true but was never able to prove or disprove it. Cantor called the hypothesis c = ℵ 1 the continuum hypothesis and went to his death not resolving the question. A proof of the continuum hypothesis would confirm that the continuum c is the smallest uncountable set and would bridge the gap between the countable and uncountable. After Cantor s death, due to the paradoxes of Bertrand Russell and others, various logicians, such as Ernst Zermelo and Abraham Fraenkel, placed the study of sets on a firm foundation with the introduction of a set of axioms. In 1938 under the framework of these axioms, the Austrian logician Kurt Godel ( ) proved that the continuum hypothesis can safely be assumed true since he proved it can t be proven false 3. This means that it can be added as an axiom to the existing axioms of set theory without introducing any inconsistencies that were not already present. Although this discovery was significant, it was not the last word since he didn t prove the hypothesis was true, nor did he prove it was independent of the axioms of ZermeloFraenkel (ZF). This later question was resolved in 1963 by the American logician Paul J. Cohen who proved that the continuum hypothesis is independent of the axioms of set theory, the net result being that Cantor s continuum hypothesis c = ℵ 1 is undecidable, meaning that anyone can accept it as true or accept it as false. If taken as a true axiom (along with the ZermeloFraenkel axioms), the resulting theory is called Cantorian set theory, while if assumed false, the theory is called noncantorian set theory). In either case, one has a valid set of axioms, albeit much different. Very strange indeed 4! The generalized continuum hypothesis, also proposed by Cantor, has piqued the interest of set theorists and logicians for 1 years. It states that for any infinite set A there is no infinite set whose cardinality is between the cardinality of A and the cardinality of its power set P ( A ). Symbolically, it states ℵα ℵ + 1 = 2, α = 1,2,... α Margin Note: Cantor s theory does test one s intuition, knowing his theory is able to prove that a line segment a thousand light years in length contains the 3 In other words consistent with the Zermelo Fraenkel axioms. 4 Most logicians accept Cantorian set theory..
6 6 same number of points, not one more or one less, than a line the width of an electron. Hmmmmmm Since there are a finite number of letters in the alphabet, one can show the number of possible names for things is denumerable. But there are sets that are uncountable, which means there are sets which cannot be named. Example 2 (Cardinality of Sequences) Solution Show that the set of all sequences of s and 1 s has cardinality c. Consider two the typical sequences of s and 1 s Sequence 1: ( ) Sequence 2: ( ) where the dots at the end of the sequences mean all remaining members of the sequences are zero. In other words Sequence 1 has 1 s in the 1 st, 4 th, and 6 th position and zeros elsewhere. Sequence 2 has 1 s in the 3 rd, 5 th, 6 th, and 9 th positions and zeros elsewhere. Taking note of the positions of the 1 s, we can match any sequence of s and 1 s subsets of the natural numbers. 1,4,6 and we can For example, Sequence 1 can be matched with the set match Sequence 2 with { 3,5,6,9 }. Hence, the family of sequences of s and 1 s can be matched in a onetoone manner with the power set of N, i.e. family of all subsets of natural numbers. See Figure 1. We will see in Section 2.6 that the family of subsets of N has cardinality of the continuum c. Sequence of s and 1 s Subset of N 1,4,7 ( ) ( ) { 1,2,3,8 } ( 1... ) { 3 } (... ) OnetoOne Correspondence Figure 1 Everyday Sets Larger than the Continuum The question now occurs, are there sets familiar to the ordinary mathematician whose cardinality larger than
7 7 the cardinality of the continuum? One example is given by Theorem 3, which we state and hint at its proof. Theorem 3: The set of all realvalued functions defined on ( ) cardinality larger than c. The cardinality of this set is ℵ 2 Proof,1 has Again, the proof is by contradiction, similar to the proof of Cantor s theorem. We assume we can match every real number in (,1) with a realvalued function on (,1 ). We then construct a rogue function not on the list, which contradicts the our assumption that such a correspondence exists. Need for Axioms in Set Theory The reader should not entertain the belief that Cantor was the first mathematician to think about sets and their operations, such as union, intersection and compliment. The concept of a set has been known for centuries. What Cantor did was to introduce the formal study of sets and in particular a deep analysis of infinite sets and their many unexpected properties. Although Cantor produced many deep ideas, his interpretation of a set was intuitive. That is, to him a set was simply a collection of objects. It was Bertrand Russell who upset that view of sets with his 192 discovery, called Russell s Paradox, which showed that the naïve view of a set as any collection of objects leads to contradictions. Thus, it became clear in order to have a consistent theory of sets (no contradictions), one must formalize the study of sets with rules of the game. That is, axioms. So for the first time in the history of humankind, the most basic of all mathematical disciplines was formulated into a set of guidelines for operation. Russell s Paradox Russell s paradox is the most famous of all settheoretic paradoxes which arises in naïve set theory and motivates the need for an axiomatic foundation for set theory. The paradox was constructed by English logician Bertrand Russell ( ), who proposed a family of set R consisting of all sets who do not contain themselves. He then asked whether the family R was itself a member of R. If R R, i.e. R contains R, then we have a contradiction since R is made up of sets that do not contain themselves. On the other hand if R R, this is also a contradiction since R contains all sets that do not contain themselves which means R R. Hence, we are left with the contradictory statement R R R R Hence, we must restrict the meaning of a set from the naïve point of view which says a set is an arbitrary collection of elements.
8 8 ZermeloFraenkel Axioms The most commonly accepted axioms of set theory are the Zermelo Fraenkel 5 axioms (ZFC) axioms, developed by German logicians Ernst Zermelo ( ) and Abraham Fraenkel ( ). Zermelo 6 conceived of providing a consistent set of axioms, like those of plane geometry, which restricts the wide latitude of Cantor s interpretation of a set, thus avoiding the bad things, like Russell s paradox, but allows enough objects to be taken as sets for ordinary use in mathematics. The C in ZFC refers to the Axiom of Choice, the most controversial and debated of the ten ZFC axioms. Although there is no agreed upon names for each of the 1 axioms, and they are often written in varying notation, we have settled on the following list. ZermeloFraenkel Axioms 7 1. Existence of a Set (sets exist) There exists a set (albeit one with no elements). That is, there is a set A such that ( x)( x A) Comment: This set is called the empty set and denoted by. This axiom says that the whole theory defined by these 1 axioms is not vacuous by stating at least one set exists. 2. Axiom of Equality (equality defined) Two sets are equal if and only if they have the same members. That is for all sets A, B ( A = B) ( x)( x A x B) Comment: The Axiom of Equality (often called the Axiom of Extension) defines what it means for two sets to be equal. The first two axioms say that the empty set (guaranteed by Axiom 1) is unique. 5 There are other axioms of set theory than the ZFC axioms, such as the Von NeumannBernaysGodel axioms (which are logically equivalent to ZFC) and the MorseKelly axioms, which are stronger than ZF. 6 Zermelo published his original axioms in 198 and were modified in 1922 by Fraenkel and Skolem, and thus today are called the ZermeloFraenkel (ZF) axioms. 7 Zermelo s original axioms were stated in the language of secondorder logic: i.e. sets were quantified (like ( A),( A) ) in addition to variables. There are versions of the ZFC axioms that use only firstorder logic, but for convenience we have quantified sets in a few instances.
9 9 3. Axiom of Sets of Size One (singletons exist) For any a and b there is a a, b, that contains exactly a and b. That is set, denoted by ( a)( b)( A)( z) z { a, b} ( z = x) ( z = y) Comment: If we pick a = b then the axiom implies there exists sets with one a. Note that the existential quantifier A quantifies a set, which is element not allowed in firstorder predicate logic. 4. Axiom of Union of Sets (unions of sets exist) We have unions of sets. For any collection of sets F = { A α }, there exists a set Y A α Λ = α, called the union of all sets in F. That is ( F )( A)( x) ( x A) ( C F )( x C ) Comment: From Axioms 3 and 4, we can construct finite sets. (We are making progress.) α Λ 5. Power Set Axiom (power sets exist) set B (think power set) such that That is for any set A there exists a X A X B 6. Axiom of Infinity (infinite sets exists) There exists a set A (think infinite) such that ( A) ( x)( x A) ( x { x} ) A Comment: This axiom creates numbers A = {,1, 2,3,... } correspondence: =, 1 = { }, 2 = { 1}, 3 { 2 } conclude that the set of natural numbers exist. according to the =, and so on. From this we 7. Correct Sets Axiom (axiom that avoids Russell s Paradox) Correct Sets Axiom (axiom that avoids Russell s Paradox) If ( ) predicate and A any set, then { x A: P( x) } p x is a defines a set, i.e. the set of elements for which the predicate P ( x ) is true. The key point is that the set { x A: P( x) } is a subset of a previously defined set A, which rules out sets of the form { x : x x} which is the Russell
10 1 paradoxical set. Axiom 7 is what is called an axiom schema, which means it is really an infinite number of axioms, one for each predicate P ( x ). 8. Image of Sets are Sets The image of a set under a function is again a set. That is, if A, B are sets and f : A B is a function with domain A and codomain B, then the image f ( A) y B : y f ( x) be defined in terms of sets as the set of ordered pairs = = is a set. (A function can {( a, b) A B : f ( a) = b} 9. Axiom of Regularity (no set can be an element of itself) Every nonempty set has an element that is disjoint from the set. That is for any set ( A ) ( x) ( x A) ( x A = ) Comment: As an example if A { a, b, c} " a " where we observe a { a, b, c} =. Note { a} { a, b, c} { a} a { a, b, c} contain themselves, like A = { 1, 2,3, A} = then we can pick our element to be = but =. What this axiom accomplishes is to keep out sets that 1. Axiom of Choice Given any collection of nonoverlapping, nonempty sets, it is possible to choose one element from each set. Comments: This is an existence axiom that claims the existence of a set formed from picking one element from a collection of nonempty sets. The objection to this axiom lies in the fact that the axiom provides no rule for how the items are selected from the sets, simply that it can be done. Comments on the t Axiom of Choice A group of mathematicians are attending a buffet dinner and while they pass through the dessert line containing plates full of different kinds of cookies, one mathematician says she will appeal to the Axiom of Choice and selects one cookie from each plate. That s it, that s the Axiom of Choice. The Axiom of Choice says given a family of nonempty sets (plates of cookies) it is possible to select one member from each set. So what s the big fuss over AC? It seems so trivial. But what happens if there are an infinite number of plates, maybe uncountable? Of course if each plate has a biggest cookie, there is an obvious way to pick one cookie from each plate. Pick the biggest cookie. In
11 11 that case one doesn t have to apply the Axiom of Choice. But what if the cookies on each plate look exactly alike, then how do you go about selecting a cookie from each plate? What is your rule of selection? Can you write it on a piece of paper? Of course not. However, there is a rule and we know this since the Axiom of Choice says so. The Axiom of Choice is a pure existence axiom and for that reason it is a bone of contention for many mathematicians who prefer a constructive method for selecting the desired set. There are good consequences and bad consequences proven when adopting the Axiom of Choice (AC). A good result is the proof of Tychonoff s theorem in topology, which states that the product of an arbitrary family of compact topological spaces is compact. On the negative side, a few bad results can be proven if one adopts AC as an axiom (along with the other axioms of ZF) and one is the existence of a nonmeasurable set in measure theory. And last but not least is the rather disconcerting paradox known as the Banach ach Tarski Paradox, which states roughly that any sphere can be decomposed into a finite number of disjoint pieces and then reassembled as two spheres identical in size to the original. Of course, don t try that at home, since the procedure appeals to AC, which means it is a pure existence result with no constructive way to show how its done. That s the nature of results that apply to AC, one arrives somewhere without knowing how one got there. Recently, logicians have tried to find a compromise for AC and seek for new axioms 8 which produces good theorems but does not lead to unwanted consequences 9. Bertrand Russell s Shoe Model for AC Bertrand Russell once gave an intuitive reason why sometimes the Axiom of Choice is required and sometimes when it is not. Suppose you are supplied with an infinite number of pairs of shoes and asked to find a selection rule for picking one shoe from each pair. In this case no Axiom of Choice is required, since you can simply the left shoe (or right) from each pair. On the other, suppose you are supplied with an infinite pair of socks and asked the same question. The socks look alike so you are unable to provide a rule of selection for choosing a sock from each pair. But the Axiom of Choice comes to the rescue, which says there is such a rule, although not explicitly stated, which specifies how to select the socks. It is the non constructive aspect of the axiom that causes angst to some mathematicians and logicians. To some pure intuitionists, the word exists belongs more to religion and not mathematics. 8 See Hamilton, A.G Numbers, Sets and Axioms, Cambridge University Press, Cambridge. 9 As the American geometer Oswald Veblen once said, The test of any axiom system lies in the theorems it produces.
12 12 Axiom of Choice WellOrdering Principle When Zermelo introduced the Axiom of Choice, most mathematicians accepted it and never gave it a second thought 1. However, in 195 Zermelo proved a theorem that gave everyone second thoughts. He proved that the Axiom of Choice was equivalent to what came to be known as the Well Ordered Principle. We are getting ahead of ourselves since we don t introduce order relations until Chapter 3, but roughly a set is wellordered if there is an ordering of elements of the set in such a way that every nonempty set has N = 1,2,3,... are well a smallest element. For example the natural numbers ordered by the usual 1 < 2 < 3 < The WellOrdered Principle says that any nonempty set can be wellordered by some order relation. As an example the principle says a wellordering exists for the real numbers, although no one has ever exhibited one, which leads one to suspect the axiom as false. Although intuitively the axiom seems hard to believe, Zermelo proved it is equivalent to the Axiom of Choice, which seems obvious. It is a rather strange situation and has caused logicians a great deal of thought. 1 However, many of the foremost mathematicians of the day did object to the axiom, including measure theory analysts Henri Lebesgue and Emile Borel.
13 13 Problems 1. Prove that if we add a member to a denumerable set, we still have a denumerable set. 2. Show that the set of all sequences of s and 1 s has cardinality c. Hint: Relate each sequence of s and 1 s to a subset of natural numbers and then use Cantor s theorem. 3. Show that the set of all onetoone correspondences between N and N is c.
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