ABSTRACT SET THEORYby
THORALF A SKOLEMProfessor of Mathematics
University of Oslo
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 developed and some critical remarks added. In particular the set-theoretic relativism is emphasized as a natural consequence of the application of Löwenheim'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 Infinite 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.
1. Historical remarks. Outlines of Cantor's theory
2. Ordered sets. A theorem of Hausdorff
3. Axiomatic set theory. Axioms of Zermelo and Fraenkel
4. The well-ordering theorem 19 5. Ordinals and alephs
6. Some remarks on functions of ordinal numbers
7. On the exponentiation of alephs
8. Sets representing ordinals
9. The notions "finite" and "infinite"
10. The simple infinite sequence. Development of arithmetic
11. Some remarks on the nature of the set-theoretic axioms. The set-theoretic relativism
12. The simple theory of types
13. The theory of Quine
14. The ramified theory of types. Predicative set theory
15. Lorenzen's operative mathematics
16. Some remarks on intuitionist mathematics
17. Mathematics without quantifiers
18. The possibility of set theory based on many-valued logic
ABSTRACT SET THEORY
by Thoraff A Skolem
1. Historical remarks. Outlines of Cantor's theory
Almost 100 years ago the German mathematician Georg Cantor was studying 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 beginning 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 (1851) (Paradoxes of the Infinite), where he mentioned some of the astonishing properties of infinite sets. Already Galileo 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 investigations 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 mathematics 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 independent of our actual thinking. Perhaps an illustrating way of expressing it is to say that when we are thinking about mathematical objects we are looking at eternal pre-existing objects. It seems clear that the word "existence" 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 Platonism. 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 restriction 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 objects the set theory talks about and therefore can be taken as an object in other collections. This is indeed clear, because there are no restrictions 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.