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Infinity Comes in All Sizes

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Infinity Comes in All Sizes U.U.D.M. Project Report 2018:41 Infinity comes in all sizes Joel Nissen Examensarbete i matematik, 15 hp Handledare: Vera Koponen Examinator: Martin Herschend Oktober 2018 Department of Mathematics Uppsala University Infinity comes in all sizes Joel Nissen October 22, 2018 Introduction This report will analyze and compare different cardinal numbers. The cardinal numbers are sets of specific sizes and is often used as the measuring sticks with which we compare the sizes of other sets. Cardinals may be finite but for the most part this report will be focusing on infinite cardinals. We call the smallest infinite cardinal ! and it can easily be thought of as the set of natural numbers (= f0; 1; 2;::: g). Every element in ! can be reached by counting up to it through finitely many elements, we therefore call sets of this size countable. Furthermore we have the infinite sets with size greater than ! and they are called uncountable. One example of such a set is the set of real numbers R. We usually use the symbol c to denote the cardinal corresponding to the size of R. However, this rises the question, is there a cardinal number κ with the size ! < κ < c? If so, what kind of set would that correspond to? The answer to the first question is that neither the existence nor the absence of such a cardinal follows from our usual axioms (ZFC). The statement that no such set exists is known as the continuum hypothesis. As this report is mostly about answering the second question we will not be assuming the continuum hypothesis. The cardinals that we will discuss will be • The smallest uncountable cardinal !1 • The pseudo-intersection number p • The dominating number d • The bounding number b • The splitting number s • The reaping number r • The almost disjoint number a • The independence number i • The homogeneity number hom • The partition number par 1 • The shattering number h • The continuum c The order in which this will be discussed will be the same as the order in the book 'Combinatorial Set Theory' by Lorenz J. Halbeisen. That is also the book that most of this report is based on. The final result will be a partial ordering of all these numbers as the ordering of these numbers is at least partially independent of ZFC, see page 193 in [Halbeisen, 2012]. This graph illustrates this partial ordering: a !1 p h par b r hom c s d i Here a line between elements means that the leftmost element is smaller than or equal to the rightmost. Basic definitions The reader of this report is assumed to have some familiarity with set theory like the basics found in [Cunningham, 2016]. Some concepts may, however, need a refresher and some key parts may be new even to a reader familiar with many of the concepts. We start by defining a well ordering. Definition 1. A well ordering of a set is a linear order where every non-empty subset has a smallest element. A common example of a well ordered set is the natural numbers N. It is worth commenting that if we assume the axiom of choice, as we will, it follows that all sets can be well ordered. Next we need Definition 2. A transitive set A is a set with the property that if a 2 A then we have a ⊂ A And with this we can define the main objects of this report, ordinals, and cardinals. Definition 3. An ordinal is a transitive set that is well ordered by the '2' relation. As the ordinals are said to be well ordered by the '2' relation it is common to use the notation κ < η to denote κ 2 η. Furthermore it can be noted that for any two ordinals κ, η we have that κ < η, κ = η or κ > η. To exemplify we have that the smallest ordinal 2 is the empty set ; which is followed by f;g < f;; f;gg and so on. One often choose to denote ; by 0, f;g by 1 and so on. This gives an intuitive picture of how the finite ordinals and the natural numbers relates to each other. Definition 4. A cardinal (or cardinal number) is an ordinal for which there exist no injective function to a smaller ordinal. A well known result is that for each set A there is a bijection to a unique cardinal, this cardinal is called the cardinality of A and is denoted jAj. This is a common way of determining the size of a set and is an important notion in several branches of mathe- matics, not the least analysis. It is clear that if there is a bijection between two sets, then they have the same cardinality. In fact an important result worth mentioning is the Schr¨oder-Berstein theorem that states that Theorem 1 (Schr¨oder-Berstein). If we have two sets A and B such that jAj ≤ jBj and jAj ≥ jBj, then jAj = jBj For proof, see page 135 in [Cunningham, 2016]. Here it may be worth mentioning that not all ordinals and cardinals are finite, indeed this report will focus on infinite car- dinals. The smallest infinite ordinal is called ! and sets with cardinality ! = f0; 1; 2; :::g are called countable and conversely sets of cardinality greater than ! are called uncount- able. We also need to establish some more notation. For two cardinals κ, λ we have that κ + λ denotes the cardinality of the set κ × f0g [ λ × f1g and κ · λ = jκ × λj. It may be worth noting that for infinite cardinals κ and λ we have κ · λ = max(κ, λ). Finally, AB denotes the set of functions from A to B. An important case of this is the set of functions from a set A to the set 2 = f;; f;gg (denoted A2). This is because of the relation between A2 and the set of all subsets of A (denoted P(A)). For every set a 2 P (A) we have that A −1 there is an unique function fa 2 2 such that fa (1) = a. That fa is unique in this way is clear since if we assume that there is a function g 2 A2 such that g−1(1) = a we note −1 −1 that fa (0) = g (0) and thus they map all elements the same and we have fa = g. A We now take the function : P (A) ! 2 such that (a) = fa. That this function is bijective has already been established. That means that we have a bijection from P (A) to A2 and thus they must have the same cardinality. Another important fact is that for an infinite cardinal κ we have that jκ2j = jκκj. For proof, see Lemma 9.2.11 on page 222 in [Cunningham, 2016]. For a set A and a cardinal κ we will also use the notation [A]κ to denote the set of all subsets of A that have cardinality κ and fin(A) is the set of all finite subsets of A.A useful result here is Lemma 1. Let A be a set of sets and let κ be a cardinal. If X ≤ κ for all X 2 A then j S Aj ≤ jAj · κ For the proof of this, see Theorem 9.2.12 on page 223 in [Cunningham, 2016]. Fur- thermore we will often call a set of sets or functions a family. For a function f : A ! B 3 and a subset S ⊂ A we use the notation fjS to denote the unique function from S to B such that it coincides with f on S. We say that fjS is the restriction of f to S. The cardinals c and !1 Another important cardinal is the so called continuum c. This is the cardinality of the real numbers and we have that ! < c. We even have c = j!!j. ! ! Proof. To prove that c = j 2j we use that c = jRj and j 2j = jN2j = jQ2j = jP (Q)j so ! c = j 2j , jRj = jP (Q)j ! First of we prove c ≤ j 2j by taking an function f : R ! P (Q) that sends each real number as follows x 7! fq 2 Q : q < xg. This is clearly injective since there is rational number between every real number. Thus c ≤ j!2j. To show the converse we construct the function g : P (N) ! R That acts like: X g(A) = 10−n n2N For each set this generates a number consisting of only ones and zeros. Assume that A; B 2 P (N) are different sets. Then there is a smallest element k that separates them so that in the kth position in the decimal expansion the generated numbers will differ, thus g(A) 6= g(B) and by the Schr¨oder-Bersteintheorem we have c = j!2j. Then, by Cunningham, we know that c = j!!j. From our usual axioms (ZFC) one can neither prove nor disprove that there is a cardinal number between ! and c. To assume that c is the smallest infinite cardinal greater than ! is called the continuum hypothesis. The cardinals that will be discussed in this report will have c as an upper bound so to assume the continuum hypothesis would render the report rather short. Instead we say that the smallest uncountable cardinal is called !1. This will be proven to be a lower bound of the rest of the cardinals in the report. The pseudo-intersection number p We define the property that a set A is almost contained in a set B if AnB is finite (denoted A ⊆∗ B).
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