For those of you new to abstract mathematics elementary does not mean simple though much of the material. The study of these topics is, in itself, a formidable task. Introduction to logic and set theory202014 general course notes ahuva c. The bibliography, the section on notation, and the index have been considerably expanded as well. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. A book of set theory, first published by dover publications, inc. Further, we call attention to the bibliographical notes which appear at the end. In mathematics, the notion of a set is a primitive notion. A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. Schwartz march 27, 2009 click here to view program. By a wellfounded set, we mean a set ssuch that every sequence s s 0 3s 1 3s 2 3is.
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. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. They are not guaranteed to be comprehensive of the material covered in the course. Oleary, the structure of proof with logic and set theory, 2002. Though propositional logic will prove a useful tool to describe certain aspects of meaning, like the reasoning in 1, it is a blunt. 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. 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. We do need to assert though that the universe of our theory contains some objects. No student is expected to really benefit from these notes unless they have regularly attended the lectures.
This note is an introduction to the zermelofraenkel set theory with choice zfc. Successive chapters examine the recursive functions, the axiom of choice, ordinal and. Set theory \a set is a many that allows itself to be thought of as a one. Center for the mathematics of uncertainty an introduction to the mathematics of uncertainty including set theory, logic, probability, fuzzy sets, rough sets, and evidence theory. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with.
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. Schwartz january 9, 1930 march 2, 2009 a celebration of the life of jacob t. The order of the elements in a set doesnt contribute. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is. 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.
This chapter will be devoted to understanding set theory, relations, functions. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Introduction to fuzzy logic, by franck dernoncourt home page email page 2 of20 a tip at the end of a meal in a restaurant, depending on the quality of service and the quality of the food. Complex issues arise in set theory more than any other area of pure mathematics. 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. This book has been reprinted with the cooperation of. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. 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. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Perhaps landau hoped to induce the student to read the preface for the teacher, but not to worry about digesting. These notes are for students of math 251 as a revision workbook and are not. 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.
Suitable for all introductory mathematics undergraduates, notes on logic and set theory covers the basic concepts of logic. 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. For example, a deck of cards, every student enrolled in. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. The consistency proofs in chapter 35 are partly from kunen and partly from the author. Fuzzy set theoryand its applications, fourth edition. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. These notes were prepared as an aid to the student. Set theory is likely to be around long after most presentday programming languages have faded from memory. The book is based on lectures given at the university of cambridge and covers the basic concepts of logic. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Basic set theory a set is a many that allows itself to be thought of as a one. 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.
Basic concepts of set theory, functions and relations. 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. This new edition will certainly become a standard reference on set theory for years to come. Duplicates dont contribute anythi ng new to a set, so remove them. Fuzzy logic is not a vague logic system, but a system of logic for dealing with vague concepts. As in fuzzy set theory the set membership values can range inclusively between 0 and 1, in. It is possible to have a 11 correspondence between an in. Part ii logic and set theory based on lectures by i. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions.
Notes on logic and set theory cambridge mathematical. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Set theory and logic supplementary materials math 103. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Fuzzy logic applications fuzzy set and fuzzy operations c 2001 by yu hen hu 3 intro. Lectures notes for an advanced course given in esslli 2010. This is being written as a textbook for math 502, logic and set theory, and. Set theory is indivisible from logic where computer science has its roots.
Basic set theory lx 502 semantics i september 11, 2008 1. Proof, sets, and logic department of mathematics boise state. Introduction to logic and set theory 202014 bgu math. A logic of meaning and synonymy, with fritz hamm, pdf file. It would be just as good to assert the existence of any set, 9xx x. 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. We will return to sets as an object of study in chapters 4 and 5. A preliminary version of the second greek edition of notes on set theory.
These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning graduate level. 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. Applications of fuzzy set theory 9 9 fuzzy logic and approximate reasoning 141 9. Set theory for computer science university of cambridge. 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. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Shkop these notes were prepared as an aid to the student.
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