There is enough material in the text for a yearlong course at the upperundergraduate level. Sets can be constructed entirely from the empty set, as in the above examples. The study of geometry, sequences, probability, etc. The purpose of this module is to introduce language for talking about sets, and some. For shorter onesemester or onequarter courses, a variety of arrangements of topics are possible. If you get a number larger than 11 or smaller than 0, add or subtract 12 to get a valid pitch class number.
The usual set theory of zermelo and fraenkel is not entirely neglected. Basic set theory a set is a many that allows itself to be thought of as a one. Hardegree 18 basic set theory, summary up to 7, set. The concept of set serves as a fundamental part of the present day mathematics. To compile the source code into a pdf yourself, you need latex. Set theory for computer science university of cambridge. Weiss university of toronto these notes for a graduate course in set theory cover the axioms of set theory, the natural numbers, the ordinal numbers, relations and orderings, cardinality, the real numbers, the universe, reflection, elementary submodels, and constructibility. The basic strategy is to reduce classical arithmetic thought of as the theory of the natural. Download the project an introduction to symbolic logic as a pdf file ready for classroom. Jul 23, 2014 a historical introduction presents a brief account of the growth of set theory, with special emphasis on problems that led to the development of the various systems of axiomatic set theory. Basic set theory gary hardegree department of philosophy university of massachusetts amherst, ma. In pure set theory, every element of every set is itself a set, so no simpleminded syntactic.
Fundamental to set theory is the notion of sets have members, also. Pure and impure sets sets can be constructed entirely from the empty set, as in the above examples. A revised and corrected republication of set theory, originally published in 1971 by addisonwesley publishing. Set theory is indivisible from logic where computer science has its roots. There is a study of somewhat more advanced topics in set theory at the. Jan 01, 2015 if you think childlike, youll stay young. S empty set in virtue of the principle of extensionality, there can be at most one set with no members. Set theory by gary hardegree free book at ebooks directory. A book of set theory, first published by dover publications, inc. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Logic is a branch of science that studies correct forms of reasoning. Hardegree, basic set theory page 4 of 39 39 next, we note that set theory officially postulates that a, a,b, a,b,c, etc.
This book has been reprinted with the cooperation of kyung moon publishers, south korea. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. The definition makes clear that for the so constructed new type of fuzzy sets i. Other logic textbooks openlogicprojectopenlogic wiki github. This chapter introduces set theory, mathematical in. This online version, which contains revisions and corrections from the printed version, is made available. This is a key part of the general program to reduce mathematics to set theory. This alone assures the subject of a place prominent in human culture. Set theory is the axiomatization of mathematics, and without ac noone seriously doubts its truth, or at least consistency. Or they can be constructed from an underlying universe of urelements,4 which are presumed not to be sets, as in the following from earlier. Basic to set theory is the notion of membership or elementhood.
Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Set theory \a set is a many that allows itself to be thought of as a one. Sets are used to define the concepts of relations and functions. Today this concept is being used in almost every branch of mathematics. Antony eagle, elements of deductive logic, github source code cc bysa 4. Then by the axiom schema of comprehension, there is a set bdf x2vw g. Hardegree, set theory an overview 1 of 34 set theory. Rather, logic is a nonempirical science like mathematics.
Georg cantor in the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains in predicate logic form sets as well. B for example, if s is the set of all juices in the supermarket, and t is the set of all. Jan 28, 2014 applications of set theory slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Set theory if we are interested in elements of a set a that are not contained in a set b, we can write this set as a. To express the relation of membership, we use a stylized epsilon symbol.
All content, except where otherwise noted, is provided under a creative commons 4. Usually we denote sets with uppercase letters, elements with lowercase letters. An introduction to modal logic 2009 formosan summer school on logic, language, and computation 29 june10 july, 2009. Basic material on set theory overview summary, basic concepts, relations, functions, natural numbers, cardinal numbers. The accompanying latex tutorial will guide you through the process of installing it on your computer, and show you the basics of how to prepare a document in. But most people do things without energy, and they atrophy their mind as well as their body. Hardegree, the conditional in abstract and concrete quantum logic. Open source means the source files from which pdfs can be. Review of set theory michael williams last updated. The following notation is used to show set membership means that x is a member of the set a means that x is not a member of the set a. The degree of success that has been achieved in this development, as well as the present stature of set theory, has been well expressed in the nicolas bourbaki.
For those of you new to abstract mathematics elementary does not mean simple though much of the material is fairly simple. The remaining notions can all be defined in terms of membership using the standard machinery of firstorder logic. But even more, set theory is the milieu in which mathematics takes place today. This book is dedicated to prabir roy, louis mcauley, jonathan. Indeed, set theory can be formulated in its entirety as a formal theory in which the only nonlogical notion symbol is membership. Basic intuitionistic principles in fuzzy set theories and its extensions.
The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via selfstudy. The theory of fuzzy sets offers natural, from the first glance, operations. Further instructions on compiling pdfs from the latex source can be found in the wiki. Compare this to the concept of modes of limited transposition found in messiaens theory. Specifically, set theory postulates the existence of a set with no members. Then the axiomatic theory of zermelofraenkel is developed. Weiss university of toronto these notes for a graduate course in set theory cover the axioms of set theory, the natural numbers, the ordinal numbers, relations and orderings, cardinality, the. Introduction this book is one of the most assign textbooks for an introductory course in set theory. They are not guaranteed to be comprehensive of the material covered in the course. An introduction to set theory university of toronto.
Membership basic to set theory is the notion of membership or elementhood. After some historical remarks the chief ideas of the naive set theory are explained. However, this is not to suggest that logic is an empirical i. Cardinal numbers page 4 of 14 14 we are now in a position, finally, to define na, at least in the finite case. Subsequent chapters explore classes and sets, functions, relations, partially ordered classes, and the axiom of choice. In mathematics, the notion of a set is a primitive notion. Asymptotic differential algebra and model theory of transseries. In the present section, we discuss the remaining ones. If you keep your energy going, and do everything with a little flair, youre gunna stay young. Numerals and numbers the next topic we consider is the set theoretic reconstruction of the theory of natural numbers. Set theory by gary hardegree download link ebooks directory. The empty set we said earlier that sets have members, but there is an exception. An introduction to elementary set theory guram bezhanishvili and eachan landreth 1 introduction in this project we will learn elementary set theory from the original historical sources by two key gures in the development of set theory, georg cantor 18451918 and richard dedekind 18311916. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin.
Set theory 1 set theory one of the main objectives of a statistician is to draw conclusions about a population of objects by conducting an experiment. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. We now in troduce the operations used to manipulate sets, using the. For those of you new to abstract mathematics elementary does not mean simple though much of the material. To transpose a set down by n half steps, subtract n from each pitch class in the set.
An introduction to independence proofs by kenneth kunen, naive set theory by paul r. Chapter 4 set theory nanyang technological university. March 30, 2009 basic set theory we will use the standard notation for containments. Skolem university of notre dame, 1962 the book contains a series of lectures on abstract set theory given at the university of notre dame. Inversional symmetry this is a property of a set which is able to map onto itself through inversion at some value of n. The integers are the set of whole numbers, both pos itive and negative. Inversion, composition, and restriction functions are a species of relation, so the various notions that apply generally to relations apply also to functions.
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