This note is an introduction to the zermelofraenkel set theory with choice zfc. Though propositional logic will prove a useful tool to describe certain aspects of meaning, like the reasoning in 1, it is a blunt. Basic concepts of set theory, functions and relations. Subsets a set a is a subset of a set b iff every element of a is also an element of b. Schwartz march 27, 2009 click here to view program.
Basic set theory lx 502 semantics i september 11, 2008 1. This book has been reprinted with the cooperation of. Suitable for all introductory mathematics undergraduates, notes on logic and set theory covers the basic concepts of logic. Proof, sets, and logic department of mathematics boise state. No student is expected to really benefit from these notes unless they have regularly attended the lectures. Set theory and logic supplementary materials math 103. A preliminary version of the second greek edition of notes on set theory. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Two sets are equal if and only if they have the same elements. The strong tradition, universality and neutrality of set theory make it rm common ground on which to provide uni cation between seemingly disparate areas and notations of computer science. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with. Motivation when you start reading these notes, the first thing you should be asking yourselves is what is set theory and why is it relevant.
By a wellfounded set, we mean a set ssuch that every sequence s s 0 3s 1 3s 2 3is. Lectures notes for an advanced course given in esslli 2010. This chapter will be devoted to understanding set theory, relations, functions. It assumes no knowledge of logic, and no knowledge of set theory beyond the vague familiarity with curly brackets, union and intersection usually expected of an advanced mathematics.
Notes on logic and set theory cambridge mathematical. Fuzzy logic is not a vague logic system, but a system of logic for dealing with vague concepts. A book of set theory, first published by dover publications, inc. 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. In an axiomatic treatment of set theory as in the book by halmos all assertions about sets below are proved from a few simple axioms. Perhaps landau hoped to induce the student to read the preface for the teacher, but not to worry about digesting. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. In naive set theory, all sets are essentially defined to be subsets of. Schwartz january 9, 1930 march 2, 2009 a celebration of the life of jacob t. Fuzzy logic applications fuzzy set and fuzzy operations c 2001 by yu hen hu 3 intro. It is possible to have a 11 correspondence between an in.
These notes are for students of math 251 as a revision workbook and are not. Applying formalized logic to analysis a biographical memoir written for the national academy of sciences by martin davis and edmond schonberg pdf file jacob t. Like logic, the subject of sets is rich and interesting for its own sake. The proof that p t in chapter 34 is based upon notes of fremlin and a thesis of roccasalvo. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Leader, lentterm 2005, 2010 chapter 1 propositional logic 1 chapter 2 wellorderings and ordinals 7 chapter 3 posets and zorns lemma 16 chapter 4 predicate logic 24 chapter 5 set theory 34 chapter 6 cardinals 43. Basic set theory a set is a many that allows itself to be thought of as a one. The study of these topics is, in itself, a formidable task. It would be just as good to assert the existence of any set, 9xx x. Set theory is indivisible from logic where computer science has its roots. Duplicates dont contribute anythi ng new to a set, so remove them. For example, a deck of cards, every student enrolled in. 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. The consistency proofs in chapter 35 are partly from kunen and partly from the author. For those of you new to abstract mathematics elementary does not mean simple though much of the material. Successive chapters examine the recursive functions, the axiom of choice, ordinal and. Leader notes taken by dexter chua lent 2015 these notes are not endorsed by the lecturers, and i have modi ed them often signi cantly after lectures. Set theory is likely to be around long after most presentday programming languages have faded from memory. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. This new edition will certainly become a standard reference on set theory for years to come. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely.
The bibliography, the section on notation, and the index have been considerably expanded as well. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. The upshot of this discussion is that the paradoxes of set theory give no grounds for doubting the consistency of the theory of trans nite numbers or its embodiment in ordinary set theory. Logic and set theory a rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. For our purposes, it will sufce to approach basic logical concepts informally. Introduction to logic and set theory 202014 bgu math. An overview of logic, proofs, set theory, and functions. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning graduate level. This is being written as a textbook for math 502, logic and set theory, and.
Complex issues arise in set theory more than any other area of pure mathematics. Set theory \a set is a many that allows itself to be thought of as a one. Introduction to logic and set theory202014 general course notes ahuva c. We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. The existence of the empty set would then follow from comprehension below. Free set theory books download ebooks online textbooks. A logic of meaning and synonymy, with fritz hamm, pdf file. These notes were prepared as an aid to the student. Fuzzy set theoryand its applications, fourth edition.
They are not guaranteed to be comprehensive of the material covered in the course. The order of the elements in a set doesnt contribute. The book is based on lectures given at the university of cambridge and covers the basic concepts of logic. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Shkop these notes were prepared as an aid to the student. Oleary, the structure of proof with logic and set theory, 2002. As in fuzzy set theory the set membership values can range inclusively between 0 and 1, in. Applications of fuzzy set theory 9 9 fuzzy logic and approximate reasoning 141 9. A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. The axioms of set theory, ordinal and cardinal arithmetic, the axiom of foundation, relativisation, absoluteness, and reflection, ordinal definable sets and inner models of set theory, the constructible universe l cohens method of forcing, independence. Leader, lentterm 2005, 2010 chapter 1 propositional logic 1 chapter 2 wellorderings and ordinals 7 chapter 3 posets and zorns lemma 16 chapter 4 predicate logic 24 chapter 5 set theory 34 chapter 6 cardinals 43 bonus lecture incompleteness examples sheets prerequisites.
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