Set theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. It was first presented in ne edward nelson, internal set theory. It is used as a foundation for many subfields of mathematics. There is a study of somewhat more advanced topics in set theory at the. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. This book is dedicated to prabir roy, louis mcauley. Today this concept is being used in almost every branch of mathematics.
I offer no definition of what a set is beyond the intuitive notion described above. This document contains the mathematical foundation of set theory. 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. The axiom of pair, the axiom of union, and the axiom of.
Because the fundamentals of set theory are known to all mathematicians, basic problems in the subject seem elementary. Starting from zfc, the exposition in this rst part includes relation and order theory as well. 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. Set theory deals with objects and their collections. The purpose of this module is to introduce language for talking about sets, and some. We explain the notions of primitive concepts and axioms. Set theory for computer science university of cambridge. In this case we also say that acontains b, and we write b. Set theory, branch of mathematics that deals with the properties of welldefined collections of objects such as numbers or functions.
There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. Set theory a mathematical model that we will use often is that of. The present notes do give all of the axioms2 of zfc. The study of geometry, sequences, probability, etc. Of the usual socalled zermelofraenkel axioms with choice, the notes did mention the axioms of foundation, in nity and choice, but not explicitly the others. Basic concepts of set theory, functions and relations. If aand bare sets, we say that bis a subset of aif every element of b is an element of a. In the areas pertaining to statistics, it is particularly used in probability. A new ap proach to nonstandard analysis, bulletin american mathematical society 83 1977.
Usually we denote sets with uppercase letters, elements with lowercase letters. An introduction to independence proofs by kenneth kunen, naive set theory by paul r. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Number theory, algebra, analysis an all other theories could be constructed within. After mathematical logic has provided us with the methods of reasoning we start with a very basic theory. Advocates of the fast development claim at least two. Philosophy of set theory university of california, irvine. Texts in set theory frequently develop the subject rapidly moving from key result to key result and suppressing many details. Tuples, families a tuple or ntuple, for any integer n is an interpretation of a list of nvariables. This view, from inside set theory and about itself, serves to shift the focus to handbook of the philosophy of science. This accessible approach to set theory for upperlevel undergraduates poses rigorous but simple arguments. This instructional unit on sets provide a stepbystep introduction to sets and set theory.
Complex issues arise in set theory more than any other area of pure mathematics. They look like they could appear on a homework assignment in an undergraduate course. Set theory and topology an introduction to the foundations of analysis 1 part i. Set theory is the axiomatization of mathematics, and without ac noone seriously doubts its truth, or at least consistency. A revised and corrected republication of set theory, originally published in 1971 by addisonwesley publishing company, reading, massachusetts.
Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. The story of mathematics, ian stewart describes the situation in this way. Axiomatic set theory january 14, 20 1 introduction one of our main aims in this course is to prove the following. August 19, 2019 1 introduction the fundamental ideas of set theory and the algebra of sets are probably the most important concepts across all areas of mathematics other than the algebra of real numbers. One of the most beautiful sources of settheoretic wisdom is still hausdorffs set theory. We write x2ato mean that xis an element of a set a, we also say that xbelongs to aor that xis in a. The semantics of predicate logic is defined in terms of set theory. In 4, 5, mololdtsov successfully applied the soft set theory in in this present paper the concept of soft. For example, the set containing only the empty set is a nonempty pure set.
Set theory is indivisible from logic where computer science has its roots. The settheory in those notes was somewhat na ve, that is, nonaxiomatic. A 2tuple is called an oriented pair, a 3tuple is a triple, a 4tuple is a quadrupletuples of a given kind list of variables with their types. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is, but we can give an informal description, describe. In contrast to naive set theory, the attitude adopted in an axiomatic development of set theory is that it is not necessary to know what the things are that are called sets or what the relation of membership means. For those of you new to abstract mathematics elementary does not mean simple though much of the material. The formal axiomatic system of ordinary set theory zfc. The concept of set serves as a fundamental part of the present day mathematics. A little set theory never hurt anybody matthew saltzman department of mathematical sciences clemson university draft. Instead, i am going to show you what can be done with sets. Basic definitions and notation, types of sets, equality, and venn diagrams are presented. As mathematicians, we often read a nice new proof of a known theorem, enjoy the different approach, but.
This unit also covers subsets, the universal set, setbuilder notation, complement, intersection and union. It is thus a metafunction from a nite metaset, to the universe. 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. The mathematical theory of sets is both a foundation in some sense for classical mathematics and a branch of mathematics in its own right. They are not guaranteed to be comprehensive of the material covered in the course. Indeed, one way to state the axioms of probability involves set theory.
Here are three simple statements about sets and functions. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Of sole concern are the properties assumed about sets and the membership relation. These will be the only primitive concepts in our system. Sets, relations, numbers felix nagel abstract we provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. B for example, if s is the set of all juices in the supermarket, and t is the set of all. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. The present book covers each of these areas, giving the reader an understanding of the ideas involved.
Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. In this section we discuss axiomatic systems in mathematics. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. Pdf axiomatic set theory download full pdf book download. A recent and highly readable addition to the literature, with. Both its foundational role and its particular mathematical features the centrality of axiomatization and the prevalence of. The axioms of set theory of my title are the axioms of zermelofraenkel set theory, usually thought ofas arisingfromthe endeavourtoaxiomatise the cumulative hierarchy concept of set. Let us consider a few examples of entities we may consider to be sets.
Set theory sets a set is a collection of objects, called its elements. In mathematics, the notion of a set is a primitive notion. Basic set theory a set is a many that allows itself to be thought of as a one. In firstorder logic, any expression is ensured to take a definite value, for every data of a model and values of all free variables there by virtue of its syntactic correction, that is implicit in the concept of expression. Sets are used to define the concepts of relations and functions. The usual set theory of zermelo and fraenkel is not entirely neglected.
Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. We declare as primitive concepts of set theory the words class, set and belong to. Set theory is a branch of mathematical logic that studies sets, which informally are collections of. Second, every mathematical statement or proof could be cast into formulas within set theory. But in set theory, this may still depend on the values of free variables. For any two sets x and y, either there is a onetoone function from. Chapter 4 set theory \a set is a many that allows itself to be thought of as a one. Much of the concepts in probability are derived from the consequences of set theory. A set is pure if all of its members are sets, all members of its members are sets, and so on. A solid foundation on sets is provided for students of all ages. 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.
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