Suangely, it was his work in the highly technical field of trigonometric series which first led Cantor to study the properties of sets. At first, he confined him- self to certain particular sets of real numbers which occurred in connection with the convergence of series, But Cantor was quick to understand that his discoveries applied to sets quite generally; in a series of remarkable papers, published between and , he moved progressively further from the concrete problems which had initiated his thinking on sets, and toward the powerful general concepts which underlie set theory today.
It was received at first with skepticism, sometimes even with open hostility. And before the tum of the century, even the most revolutionary aspects of set theory had been accepted by a great many mathematicians—chiefly because they turned out to be invaluable tools, particularly in analysis, Meanwhile,. Many of the ancient schools, from Euclid through the Middle Ages, contended that the various branches of mathematics could be subsumed under geometry numbers might be conceived as geometric propor- tions ; a far more successful attempt at unification came in the nineteenth century, when the work of Weierstrass, Dedekind, and others suggested that all of classical mathematics could be derived from the arithmetic of the natural numbers positive integers.
All the proper- ties of the natural numbers can be proven using these definitions and elementary set theory. These paradoxes had such wide repercussions that it is worth looking at them in-some detail. The earliest of the paradoxes was published in by Burali-Forti, but it had already been discovered, two years earlier, by Cantor himself, Since the Burali-Forti paradox appeared in a rather technical region of set theory, it was hoped, at first, that a slight alteration of the basic definitions would be sufficient to correct it.
However, in Bertrand Russell gave a version of the paradox which involved the most elementary aspects of set theory, and therefore could not be ignored. We will devote the remainder of this section to the presentation of two of the most celebrated paradoxes, which involve only elementary concepts of set theory.
Now the possibility arises that A may be an element of itsclf; for example, the set of all sets has this property. Thus, we have proven that Sis an element of S if and only if S is not an element S—a contradiction of the most fundamental sort.
Usually, in mathematics, when we reach a contradiction of this kind, we are forced to admit that one of our assumptions was in error. Since there are only a finite number of English words, there are only finitely many combiriations of fewer than twenty such words—that is, T is a finite set.
Quite obviously, then, there are natural numbers which are greater than all the elements of 7; hence there is a least natural number which cannot be described in fewer than twenty words of the English language.
By definition, this number is not in T; yet we have described it in sixteen words, hence it is in. Before the paradoxes, the question of the existence of sets had never been posed. In the various movements which sprang up, during the early 's, with the aim of revising the foundations of set theory, the topic of central concern was the existence of sets. What properties legitimately defined sets? Under what conditions do properties define sets at all? How can new sets be formed from existing ones?
The remainder of this chapter is devoted to presenting these three ways of thought. First, however, we shall briefly review the development of the axiomatic method. It is worth examining a few of the major developments in geometry which influenced the growth of the axiomatic method.
Perhaps the greatest defect in the Elements is the number of tacit assumptions made by Fuclid—assumptions not granted by the postulates.
For example, in a certain proof it is assumed that two circles, each passing through the center of the other, have a pair of points in common—yet the postulates do not provide for the existence of these points. Other arguments in the Elements involve the concept of rigid motion—a concept which is not defined or mentioned in the postulates.
Thus, throughout Buclid, the orderly chain of logical inferences is frequently broken by tacit appeals to visual evidence. Pasch published the first formulation of geometry in which the exclusion of any appeal to intuition is clearly stated as a goal and systematically carried out. By the end of the nineteenth century, then, a modern conception of the axiomatic method began to emerge.
However, there was a new understanding of the formal nature of mathematical proof, Inasmuch as possible, the axioms should be sufficiently detailed, and the rules of logical deduction sufficiently explicit, that neither intuition nor intelligence is needed to go through the steps of a proof.
Ideally, it should be possible for a computer to verify whether or not a proof is correct. As long as mathematics is formulated in ordinary languages, such as English, human understanding is indispensable for interpreting statements and finding the structure of complex sentences. The creation of formal, symbolic languages was one of the most important developments of modern mathematics; here is what such a language looks like.
It is convenient to denote a predicate by a single letter followed by the list of its variables. Itis a remarkable fact that every known branch of mathematics requires only a finite number usually a very small number of distinct elementary predicates. BOx, y, 2 : y is between x and z. C u, x,y : the segment wv is congruent to the segment xy.
Du,v,w,x,y, 2 : the angle wow is congruent to the angle xyz. PAQ: PandQ. PvQ: Porg. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. New post summary designs on greatest hits now, everywhere else eventually. Related 2. Hot Network Questions. Question feed. ISBN pbk. ISBN 0. Set theory. Title, QA P55 Historical Introduction The background of set theory. Axiomatic set'theory Objections to the axiomatic proach.
Generalized union and intersection SO. Properties of denumerable sets Arithmetic of Cardinal Numbers Introduction Bibliography 6. Tadopted this guiding principle as a categorical imperative during the writing of this book. Fven afterall the mathematics was in place, [reread and rewrote many times and tested explanatory strategies with my students, [have dedicated this book to them because they have been my most patient and honest critics, The book owes its present form largely o them and to their sometimes naive, often brilliant suggestions, Mathematics has a superbly efficient language by means of which vast amounts of information can be elegantly expressed in a few formal definitions and theorems, Itis remarkable that the life work of consecutive generations of great thinkers can often be summed up in a set of equations, The economy of the language masks the richness and complexity of the thoughts that lie behind the symbols.
Every mathematics student has to master the conventions for using its language effectively. Now T understand! Now I see it! What is most unique about set theory is that it is the perfect amalgam of the visual and the abstract. The notions of set theory, and the ideas behind many of the proofs, present themselves to the inner eye in vivid detail.
These pictures are not as overtly visual as those of geometry or calculus. But these images are the way into abstraction. For the maturing student, the journey deeper into abstraction is a rite of passage into the heart of mathematics. Ts the continuum hypothesis a fact of the world?
Is the axiom of choice a truth? If we cannot answer with a definite yes or no, in what manner are they justified? In this book I have tried, insofar as Possible, not to evade these questions nor to dwell on them excessively. This book is a revised and re-written yersion of an earlier edition, published in by Addison-Wesley.
I have retained most of the formal definitions, theorems and proofs, with nothing more than a few corrections where needed. I have also retained the initial chapter that narrates the origins and early history of set theory, because history in general does not change, I have added commeniary, introduced some new discussions, and reorganized a few proofs in order to make them cleaner and cleatet. I have set the discussion of these topics at a level that is accessible to undergraduates while not concealing the difficulties of the subject.
From the earliest times, mathematicians haye been led to consider sets of objects of one kind or another, and the elementary notions of modern set theory are implicit in a great many classical arguments.
However, it was not until the latter part of the nineteenth century, in the work of Georg Cantor , that sets came into their own as the principal object of a mathematical theory. Suangely, it was his work in the highly technical field of trigonometric series which first led Cantor to study the properties of sets.
At first, he confined him- self to certain particular sets of real numbers which occurred in connection with the convergence of series, But Cantor was quick to understand that his discoveries applied to sets quite generally; in a series of remarkable papers, published between and , he moved progressively further from the concrete problems which had initiated his thinking on sets, and toward the powerful general concepts which underlie set theory today.
It was received at first with skepticism, sometimes even with open hostility. And before the tum of the century, even the most revolutionary aspects of set theory had been accepted by a great many mathematicians—chiefly because they turned out to be invaluable tools, particularly in analysis, Meanwhile,. Many of the ancient schools, from Euclid through the Middle Ages, contended that the various branches of mathematics could be subsumed under geometry numbers might be conceived as geometric propor- tions ; a far more successful attempt at unification came in the nineteenth century, when the work of Weierstrass, Dedekind, and others suggested that all of classical mathematics could be derived from the arithmetic of the natural numbers positive integers.
All the proper- ties of the natural numbers can be proven using these definitions and elementary set theory. These paradoxes had such wide repercussions that it is worth looking at them in-some detail. The earliest of the paradoxes was published in by Burali-Forti, but it had already been discovered, two years earlier, by Cantor himself, Since the Burali-Forti paradox appeared in a rather technical region of set theory, it was hoped, at first, that a slight alteration of the basic definitions would be sufficient to correct it.
However, in Bertrand Russell gave a version of the paradox which involved the most elementary aspects of set theory, and therefore could not be ignored. We will devote the remainder of this section to the presentation of two of the most celebrated paradoxes, which involve only elementary concepts of set theory. Now the possibility arises that A may be an element of itsclf; for example, the set of all sets has this property.
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