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. Sentential logic we go into the mathematical theory of the simplest logical notions. An elementary introduction to logic and set theory. For our purposes, it will sufce to approach basic logical concepts informally. They are not guaranteed to be comprehensive of the material covered in the course. Cambridge studies in advanced mathematics includes bibliographical references and index. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. As in fuzzy set theory the set membership values can range inclusively between 0 and 1, in. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. 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.
Oleary, the structure of proof with logic and set theory, 2002. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. For example the use of transistors instead of vacuum tubes is a paradigm shift likewise the development of fuzzy set theory from conventional bivalent set theory is a paradigm shift. Somewhat less common is the inclusion of a chapter on pointsets which focuses on results of interest to analysts and introduces the reader to the continuum problem. 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.
Part ii logic and set theory theorems based on lectures by i. 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. Subsets a set a is a subset of a set b iff every element of a is also an element of b. 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. 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. Nb note bene it is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs.
Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Basic concepts of set theory, functions and relations. The proof that p t in chapter 34 is based upon notes of fremlin and a thesis of roccasalvo. Basic set theory a set is a many that allows itself to be thought of as a one.
To discuss examples it is convenient to introduce some notation. This chapter will be devoted to understanding set theory, relations, functions. The order of the elements in a set doesnt contribute. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. When we use letters as elements of sets we do not consider.
Lecture notes on elementary logic and set theory by jeanmarc. Suitable for all introductory mathematics undergraduates, notes on logic and set theory covers the basic concepts of logic. Successive chapters examine the recursive functions, the axiom of choice, ordinal and. Set theory and logic supplementary materials math 103.
Lectures notes for an advanced course given in esslli 2010. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. By a wellfounded set, we mean a set ssuch that every sequence s s 0 3s 1 3s 2 3is. The book is based on lectures given at the university of cambridge and covers the basic concepts of logic. The study of these topics is, in itself, a formidable task. A logic of meaning and synonymy, with fritz hamm, pdf file. Publication date 1987 topics logic, symbolic and mathematical, set theory. This note is an introduction to the zermelofraenkel set theory with choice zfc. A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. 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. These notes are for students of math 251 as a revision workbook and are not meant to. Schwartz march 27, 2009 click here to view program. This book has been reprinted with the cooperation of.
If you apply t he axiom of replacement to a functionclass f, you must explain clearly why f is indeed a functionclass. Logic and set theory a rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. This is being written as a textbook for math 502, logic and set theory, and. This short textbook provides a succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Notes on logic and set theory cambridge mathematical. Schwartz january 9, 1930 march 2, 2009 a celebration of the life of jacob t. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. We cant think of examples of the use of intransitive verbs. Duplicates dont contribute anythi ng new to a set, so remove them. For those of you new to abstract mathematics elementary does not mean simple though much of the material.
State the axiom of foundation and the principle of induction, and show that they are equivalent in the presence of the other axioms of zfc. An overview of logic, proofs, set theory, and functions. It only remains to define a, b in terms of set theory. 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. Basic set theory lx 502 semantics i september 11, 2008 1. Certainly there is room for more examples and more simple exercises to. No student is expected to really benefit from these notes unless they have regularly attended the lectures. For example, a deck of cards, every student enrolled in. 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.
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. Proof, sets, and logic department of mathematics boise state. Introduction to logic and set theory 202014 bgu math. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is.
A preliminary version of the second greek edition of notes on set theory. Part ii logic and set theory based on lectures by i. Though propositional logic will prove a useful tool to describe certain aspects of meaning, like the reasoning in 1, it is a blunt. In mathematics, the notion of a set is a primitive notion. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. Two sets are equal if and only if they have the same elements. It will be suitable for all mathematics undergraduates coming to the subject for the first time. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with. Complex issues arise in set theory more than any other area of pure mathematics. Download notes on logic and set theory 97805235027. Set theory \a set is a many that allows itself to be thought of as a one. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely.
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