# pc蛋蛋 wangji: 抽象置数理论 Abstract Set Theory

NOTRE DAME MATHEMATICAL LECTURES Number 8 ABSTRACT SET THEORY by THORALF A. SKOLEM Professor of Mathematics University of Oslo, Norway NOTRE DAME, INDIANA 1962 Copyright 1962 UNIVERSITY OF NOTRE DAME COMPOSITION BY BELJAN, ANN ARBOR, MICHIGAN PHOTOLITHOPRINTED BY GUSHING - MALLOY, INC. ANN ARBOR, MICHIGAN, UNITED STATES OF AMERICA 1962 PREFACE The following pages contain a series of lectures on abstract set theory given at the University of Notre Dame during the Fall Semester 1957-58. After some historical remarks the chief ideas of Cantor s theory, now usually called the naive set theory, are explained. Then the axiomatic theory of Zermelo-Fraenkel is de- veloped and some critical remarks added. In particular the set- theoretic relativism is emphasized as a natural consequence of the application of Lowenheim s Theorem on the axioms of set theory. Other versions of axiomatic set theory which logically are of very similar character are not dealt with. However, the simple theory of types, Quine s theory and the ramified theory of types are treated to a certain extent. Also Lorenzen s operative mathematics and the intuitionist mathematics are outlined. Further, there is a short remark on the possibility of finitist mathematics in a strict sense and finally some hints are given about the possibility of a set theory based on a logic with an in- finite number of truth values. The book “Transfinite Zahlen“ by H. Bachmann has been very useful in particular for the writing of parts 6 and 8. Some references to the literature on these subjects occur scattered in the text, but no attempt has been made to set up a complete list. Such a task seems indeed scarcely worth while, because very extensive and complete lists can be found both in the mentioned book of Bachmann and in the book “Abstract Set Theory“ by A. Fraenkel. Th. Skolem. CONTENTS 1. Historical remarks. Outlines of Cantor s theory 1 2. Ordered sets. A theorem of Hausdorff 7 3. Axiomatic set theory. Axioms of Zermelo and Fraenkel 12 4. The well-ordering theorem 19 5. Ordinals and alephs 22 6. Some remarks on functions of ordinal numbers 28 7. On the exponentiation of alephs 32 8. Sets representing ordinals 35 9. The notions “finite“ and “infinite“ 38 10. The simple infinite sequence. Development of arithmetic 41 11. Some remarks on the nature of the set-theoretic axioms. The set-theoretic relativism 45 12. The simple theory of type.s 48 13. The theory of Quine 50 14. The ramified theory of types. Predicative set theory 52 15. Lorenzen s operative mathematics 61 16. Some remarks on intuitionist mathematics 64 17. Mathematics without quantifiers 68 18. The possibility of set theory based on many-valued logic 69 ABSTRACT SET THEORY fay Thoralf A. Skolem 1. Historical remarks. Outlines of Cantor s theory Almost 100 years ago the German mathematician Georg Cantor was study- ing the representation of functions of a real variable by trigonometric series. This problem interested many mathematicians at that time. Trying to extend the uniqueness of representation to functions with infinitely many singular points he was led to the notion of a derived set. This was not only the begin- ning of his study of point sets but lead him later to the creation of transfinite ordinal numbers. This again lead him to develop his general set theory. The further development of this, the different variations or modifications of it that have been proposed in more recent years, the discussions and criticisms with regard to this subject, will constitute the contents of my lectures on set theory. One ought to notice that there have been some anticipations of Cantor s theory. For example B. Bolzano wrote a paper with the title: Paradoxien des Unendlichen (1951) (Paradoxes of the Infinite), where he mentioned some of the astonishing properties of infinite sets. Already Galilei had noticed the remarkable fact that a part of an infinite set in a certain sense contained as many elements as the whole set. On the other hand it ought to be remarked that about the same time that Cantor exposed his ideas some other people were busy in developing what we today call mathematical logic. These investi- gations concerned among other things the fundamental notions and theorems of mathematics, so that they should naturally contain set theory as well as other more elementary or ordinary parts of mathematics. A part of the work of another German mathematician, R. Dedekind, was also devoted to studies of a similar kind. In particular, his book “Was sind und was sollen die Zahlen“ belongs hereto. In my following first talks I will however confine my subject to just an exposition of the most characteristic ideas in Cantor s work, mostly done in the years 1874-97. The real reason for a mathematician to develop a general set theory was of course the fact that in mathematics we often have to do not only with single mathematical objects but also with collections of them. Therefore the study of properties of such collections, even infinite ones, must be of very great importance. There is one fact to which I would like to call attention. Most of mathe- matics and perhaps above all the classical set theory has been developed in accordance with the philosophical attitude called Platonism. This standpoint means that we consider the mathematical objects as existing before and in- dependent of our actual thinking. Perhaps an illustrating way of expressing it is to say that when we are thinking about mathematical objects we are look- ing at eternal preexisting objects. It seems clear that the word “existence“ 2 LECTURES ON SET THEORY according to Platonism must have an absolute meaning so that everything we talk about shall either exist or not in a definite way. This is the philosophical background for classical mathematics generally and perhaps in particular for classical set theory. Being aware of this, Cantor explicitly cites Plato. Everybody is used to saying that a mathematical fact has been discovered, not that it has been invented. That shows our natural tendency towards Plato- nism. Whether this philosophical attitude is justified or not, however, I will not discuss now. It will be better to postpone that to a later moment. When Cantor developed his theory of sets he liked of course to conceive the notion “set“ as general as possible. He therefore desired to give a kind of definition of this notion in accordance with this most general conception. A definition in the proper sense this could not be, because a definition in the proper sense means an explanation of a notion by means of more primitive or previously defined notions. However, it is evident that the notion “set“ is too fundamental for such an explanation. Cantor says that a set is a collection of arbitrary well-defined and well-distinguished objects. What is achieved, perhaps, by this explanation is the emphasizing that there shall be no restric- tion whatever with regard to the nature of the considered objects or to the way these objects are collected into a whole. Taking the Platonist standpoint, it is clear that this whole, the collection, must itself again be considered as one of the objetts the set theory talks about and therefore can be taken as an object in other collections. This is indeed clear, because there are no re- strictions as to the nature of the objects. Now we are very well acquainted with sets in daily life. These sets are finite, but I shall not now enter into the distinction between finite and infinite sets. The most important mathematical property of the finite sets is the number of their elements. By the way I write me M, expressing that m is an element of or belongs to M. Indeed this notation is used everywhere in the literature. If we shall compare two finite sets M and N with regard to number, we may do that in the way of pairing off the ele- ments by distributing as far as possible the elements of M and N into disjoint pairs. Let us for simplicity assume M and N disjoint, that is, without com- mon elements. If it is possible to distribute the elements of M and N into dis- joint pairs (m,n), meM, neN, such that all meM and all neN occur in these pairs, then it is evident that there are just as many elements m in M as ele- ments n in N. If, however, we may build a set of pairs (m,n) such that all m occur, but not all n, then in the case of finite sets M possesses less elements than N. It is clear that it must be possible to compare sets by considering such sets of disjoint pairs in the case of infinite sets as well. This leads to one of the most important notions not only in the classical set theory but also in ordinary mathematics, namely, the notation of one-to-one correspon*- dence or mapping. We say that f is a one-to-one correspondence between the sets M and N, if f is a set of mutually disjoint pairs (m,n) such that each meM and each neN occur in one of the pairs. In order to be able to take into account the case that M and N have some common elements, it is necessary to replace the simple notion pair {a,b}, which means the set containing a and b as elements, with the notion ordered pair (a,b), which can be conceived as {{a,b}, {a} }. However I will here, to begin with, use the notion ordered pair, triple etc. as known ideas without worrying about an analysis of them. HISTORICAL REMARKS 3 Possessing the notion one-to-one correspondence or mapping, we may obtain this generalisation of the number concept: M and N have the same cardinal number, if a mapping f exists of M on N. This circumstance is written M ~ N. Cantor says that the cardinal number M of M is what remains, if we make an abstraction with regard to the individual characters of Us elements. This definition is made much clearer by Russell, who says that M is the set of all sets N being ~ M. _ Further, this definition of the relatiqg = Between cardinals was natural: M i N if M is ~ a subset of N. Further M a certain negative integer is of the same cardinality as the series of non-negative in- tegers. A little more remarkable is the fact that this is true of the set of all rational integers, negative, positive or zero. The last fact is verified by writing the integers for ex. in this order: 0, -1, 1, -2, 2, -3, 3, Or in other words, if we put for x = 0 y = 2x and for x 0, a and b coprime integers. Then we arrange the ra- tionals so that lal + b successively takes the values 1,2,3, and the for HISTORICAL REMARKS 5 which lal + b has the same value we arrange according to their magnitude. Thus we obtain the sequence ￡ l l ± l l 2 ^ j l j ^I3:^ j 4 H ^ ^z3I^ ^ 4 l H ^ + 3 j 4 1 1 1 1 2 2 I9 1 3 3 1 1 2 3 4 4 3 2 1 containing all the rational numbers. Cantor proved also that even the set of all algebraic numbers is denum- erable. This can be done in the simplest way as follows. Every algebraic number is a root in an irreducible equation anxn + + ao = 0 for some n, the a0, an being integers with 1 as g.c. div. Now we can arrange the n-tuples an, , a0 in a sequence by taking the successively increasing values of m = |an| + + Ia0l + n. Those with the same m we can take according to increasing values of n, and for those with the same value of m and n, which are only finite in number, we arrange the corresponding roots first according to their absolute value and finally those which have the same absolute value we arrange according to increasing amplitude. One might get the impression that all infinite sets were denumerable. However, Cantor proved that the set of all real numbers, even all reals be- tween 0 and 1, is not denumerable. His proof is performed by the diagonal method, called after him in the literature: Cantor s diagonal method. We know that every real number = 0 and 0. The definition of such a mapping is particularly easy when we make use of the development of reals in continued fractions. Any positive real number a can be developed thus: a1 = ao + * 6 LECTURES ON SET THEORY where a 2, the points (x,y), where x is irrational, y rational, are mapped on the irrational z such that 1 0, are mapped in an arbitrary way on the rational numbers z 0. Cantor also proved generally that the set UM of subsets of a set M was of higher cardinality than M; however, I will talk about this theorem later. ORDERED SETS. 2. Ordered sets. A theorem of Hausdorff. One obtains a more complete idea of Cantor s work by studying his theory of ordered sets. As to the notion “ordered set“ this is nowadays mostly de- fined in the following way: A set M is ordered by a set P E M2, if and only if the following state- ments are valid: 1) No pair (m,m), meM, is eP. 2) For any two different elements m and n of M either (m,n)eM or (n,m)eM but not both at the same time. 3) for all m,n,peM we have (m,n)eP and u* denote the types of the sets of positive and of negative integers. 8 LECTURES ON SET THEORY An interesting class of ordered sets are the dense ones. An ordered set is called dense, if there is always an element between two arbitrary ones. The simplest example is the set of rational numbers in their natural order. This set is also open, that means that there is no first and no last member. Now we have the remarkable theorem: There is one and only one open and dense denumerable ordertype. Proof. Let A = {ai, a2,} and B = {bi, b2,} be two denumerable sets, both open and dense. First we let ai correspond to bi. Then ai divides the remaining elements of A into those ai. Let ami be the aj with least index ai. Either mi or m2 is 2. Letting bnj be the bj with least j bi, then either ni or n2 is 2. We let bni correspond to ami and bn to am . Now every remaining a^ from A is either ami but ai but am2, which gives 4 cases. There are 4 corresponding cases for the remaining bj. Then if am3 is the a^ with least i such that aj bi we map b2 on an element a“ ai. Then b3 is either less than both bi and b2 or lies between bi and b2 or is greater than both. Respectively we map b3 on an element aflf having the same order relation to ai and aff and so on. Let us use the term scattered set for a set having no dense subset. Then an interesting theorem of Hausdorff says that every ordered set is either scattered or the sum of a set T of such sets, where T is dense. Proof: It is easy to understand that if an interval a to b in an ordered set is scattered and the interval b to c as well, then the whole interval a to c has the same property. Indeed, if d 1, ni, n2 , ., which is absurd. Similarly in the second case. Among the ordered sets, the well-ordered ones, namely those possessing a least element in every non-empty subset, are especially important. That well-ordering is equivalent to the principle of transfinite induction is well known. This principle says that if a statement S is always valid for an ele- ment of a well-ordered set M when it is valid for all predecessors, then S is valid for all elements of M. Further I ought to mention that the sum of a well-ordered set T of well-ordered sets A,B,C, is again