Introduction to Set Theory, Revised and Expanded

Karel Hrbacek, Thomas Jech

• Editorial: CRC Press
• Año: 2017
• ISBN:
  9781351990547
• Edición: 3
• Categoría: Mathematics
• Categoría: Applied Mathematics

Thoroughly revised, updated, expanded, and reorganized to serve as a primary text for mathematics courses, Introduction to Set Theory, Third Edition covers the basics: relations, functions, orderings, finite, countable, and uncountable sets, and cardinal and ordinal numbers. It also provides five additional self-contained chapters, consolidates the material on real numbers into a single updated chapter affording flexibility in course design, supplies end-of-section problems, with hints, of varying degrees of difficulty, includes new material on normal forms and Goodstein sequences, and adds important recent ideas including filters, ultrafilters, closed unbounded and stationary sets, and partitions.

Chapter 1

Sets

1. Introduction to Sets

The central concept of this book, that of a set, is, at least on the surface, extremely simple. A set is any collection, group, or conglomerate. So we have the set of all students registered at the City University of New York in February 1998, the set of all even natural numbers, the set of all points in the plane π exactly 2 inches distant from a given point P, the set of all pink elephants.

Sets are not objects of the real world, like tables or stars; they are created by our mind, not by our hands. A heap of potatoes is not a set of potatoes, the set of all molecules in a drop of water is not the same object as that drop of water. The human mind possesses thè ability to abstract, to think of a variety of different objects as being bound together by some common property, and thus to form a set of objects having that property. The property in question may be nothing more than the ability to think of these objects (as being) together. Thus there is a set consisting of exactly the numbers 2, 7, 12, 13, 29, 34, and 11,000, although it is hard to see what binds exactly those numbers together, besides the fact that we collected them together in our mind. Georg Cantor, a German mathematician who founded set theory in a series of papers published over the last three decades of the nineteenth century, expressed it as follows: “Unter einer Menge verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten in unserer Anschauung oder unseres Denkens (welche die Elemente von M genannt werden) zu einem ganzen.” [A set is a collection into a whole of definite, distinct objects of our intuition or our thought. The objects are called elements (members) of the set.]

Objects from which a given set is composed are called elements or members of that set. We also say that they belong to that set.

In this book, we want to develop the theory of sets as a foundation for other mathematical disciplines. Therefore, we are not concerned with sets of people or molecules, but only with sets of mathematical objects, such as numbers, points of space, functions, or sets. Actually, the first three concepts can be defined in set theory as sets with particular properties, and we do that in the following chapters. So the only objects with which we are concerned from now on are sets. For purposes of illustration, we talk about sets of numbers or points even before these notions are exactly defined. We do that, however, only in examples, exercises and problems, not in the main body of theory. Sets of mathematical objects are, for example:

1.1 Example

(a) The set of all prime divisors of 324.

(b) The set of all numbers divisible by 0.

(c) The set of all continuous real-valued functions on the interval [0,1].

(d) The set of all ellipses with major axis 5 and eccentricity 3.

(e) The set of all sets whose elements are natural numbers less than 20.

Examination of these and many other similar examples reveals that sets with which mathematicians work are relatively simple. They include the set of natural numbers and its various subsets (such as the set of all prime numbers), as well as sets of pairs, triples, and in general n-tuples of natural numbers. Integers and rational numbers can be defined using only such sets. Real numbers can then be defined as sets or sequences of rational numbers. Mathematical analysis deals with sets of real numbers and functions on real numbers (sets of ordered pairs of real numbers), and in some investigations, sets of functions or even sets of sets of functions are considered. But a working mathematician rarely encounters objects more complicated than that. Perhaps it is not surprising that uncritical usage of “sets” remote from “everyday experience” may lead to contradictions.

Consider for example the “set” R of all those sets which are not elements of themselves. In other words, R is a set of all sets x such that x ∉ x (e reads “belongs to,” ∉ reads “does not belong to“). Let us now ask whether R ∈ R. If R ∈ R, then R is not an element of itself (because no element of R belongs to itself), so R ∉ R; a contradiction. Therefore, necessarily R ∉ R. But then R is a set which is not an element of itself, and all such sets belong to R. We conclude that R ∈ R again, a contradiction.

The argument can be briefly summarized as follows: Define R by: x ∈ R if and only if x ∉ x. Now consider x = R; by definition of R, R ∈ R if and only if R ∉ R; a contradiction.

A few comments on this argument (due to Bertrand Russell) might be helpful. First, there is nothing wrong with R being a set of sets. Many sets whose elements are again sets are legitimately employed in mathematics — see Example 1.1 — and do not lead to contradictions. Second, it is easy to give examples of elements of R; e.g., if x is the set of all natural numbers, then x ∉ x (the set of all natural numbers is not a natural number) and so x ∈ R. Third, it is not so easy to give examples of sets which do not belong to R, but this...