Choi department of mathematical science kaist, daejeon, south korea fall semester, 2012 s. Set theory and logic thomas forster this course is the sequel to the part ii course with the title logic and set theory. No student is expected to really benefit from these notes unless they have regularly attended the lectures. B for example, if s is the set of all juices in the supermarket, and t is the set of all. Boolean algebra is given by a set s and two operations. In case we do not want to specify the target set b, we can also write f. 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. This mustread text presents the pioneering work of the late professor jacob jack t. Proof, sets, and logic department of mathematics boise state. It is free math help boards we are an online community that gives free mathematics help any time of the day about any problem, no matter what the level.
These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. 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. Choi kaist logic and set theory october 7, 2012 1 26. Logic and set theory these notes correspond to mathematical appendix 1 in the text. Unit 1 set theory and logic notes booklet unit 1 set theory and logic worksheet booklet unit 1 set theory and logic worksheet answers foundations 30 unit 1 portfolio lesson 1 types of sets and set notation lesson 1 video homework. Second, every mathematical statement or proof could be cast. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. Unlike axiomatic set theories, which are defined using a formal logic, naive set theory is defined informally, in natural language. 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. It is possible to have a 11 correspondence between an in. Set theory is likely to be around long after most presentday programming languages have faded from memory. Evidence for settheoretic truth and the hyperuniverse programme. Basic logic and set theory this short overview is designed to lay down the rudimentary aspects of sets and their associated notation. Cambridge studies in advanced mathematics includes bibliographical references and index.
Ifthereisaproofoffrom,say provesorsyntacticallyimplies,written. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Basic set theory a set is a many that allows itself to be thought of as a one. To start viewing messages, select the forum that you want to visit from the selection. Basic set theory and logic university of cambridge. The basic idea of a set, set intersection, set union etc. Like logic, the subject of sets is rich and interesting for its own sake. Few relative success storiesback to second order logicidentity crisis againassuming lst property as an axiom schemata for set theory second order logic is very set theoretical 1. But a computational logic has been formally derived from the zf axioms using isabelle.
He includes many methodological remarks and explanations, and demonstrates how the basic concepts of mathematics can be reduced to set theory. 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. The introduction to nlp module assumes some basic knowledge of set theory and logic. Paulson computer laboratory university of cambridge 4 november 1992 abstract zermelofraenkel zf set theory is widely regarded as unsuitable for automated reasoning. There are many other details and plenty of terminology that i will skip.
For example, a deck of cards, every student enrolled in. 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. Whether a given structure like the natural numbers hn. This is being written as a textbook for math 502, logic and set theory, and. Jul 27, 2006 at least universities think like you, i. An elementary introduction to logic and set theory.
Attempts to provide an axiomatic basis for mathematics were undertaken by such prominent individuals as bertrand russell, alfred north whitehead, and david hilbert. Show that every countable model of the theory of densely totally ordered sets without greatest or least elements cf. Choi kaist logic and set theory november 22, 2012 1 25. Fortunately, intuition generally does not fail us when it comes to the. 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.
Set theory \a set is a many that allows itself to be thought of as a one. Logic and set theory virginia commonwealth university. We cant think of examples of the use of intransitive verbs. Curiously, set theory arose in the context of sets of real numbers related to the convergence of fourier series. Set theory and logic is the result of a course of lectures for advanced undergraduates, developed at oberlin college for the purpose of introducing students to the conceptual foundations of mathematics. Mathematical logic is a branch of mathematics which is mainly concerned with the relationship between semantic concepts i. In this introduction to set theory and logic, the author discusses first order logic, and gives a rigorous axiomatic presentation of zermelofraenkel set theory.
Ellermeyer august 18, 2000 set theory membership a setis a welldefined collection of objects. There you develop logic for finite or explicitly presented countable vocabularies. Although it will touch most of the themes of that part ii course, and will have something of the character of a course with a title like a twentyfour lecture graduate course in logic it will concentrate. If is a set of sentences of l, then mod, the class of all models of, is the class of all interpretations of l which make all sentences in true. 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 1829, dirichlet had shown that a function always had a fourier series converging to itself, under certain sufficient conditions he considered periodic functions with finitely many extrema in every period and equal to the halfsum of. Propositions and logical operatorstautology, contradiction and contingencyrules of replacement introduction.
Ultimately, the goal of set theory was to provide a common axiomatic basis for all of mathematics. For those of you new to abstract mathematics elementary does not mean simple though much of the material. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the. Students need to be familiar with the following concepts. 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. A total order or linear order on a set is a relation logic to axiomatic set theory. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. In some sense, mathematics could then be reduced to logic. Propositional logic, interpretation with respect to a model expressed in terms of sets.
Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Set the ory deals with objects and their collections. I discuss three potential sources of evidence for truth in set theory. Using this, you can develop formal set theory, and inside set theory you can redevelop logic, and go on to model theory, etc, now without any size or definability restrictions. Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms.
Any object which is in a set is called a member of the set. Unlike axiomatic set theories, which are defined using a formal logic, naive. Lecture notes on elementary logic and set theory by jeanmarc. Introduction to logic and set theory 202014 bgu math. The standard book is paul halmos, naive set theory 1960. It only remains to define a, b in terms of set theory. Discrete mathematics, set theory, and logic discrete math, graph theory, combinatorics, recurrence relations difference equations, set theory, gene. In mathematics, the notion of a set is a primitive notion.
Set theory was developed in the second half of the nineteenth century. Set theory provides a foundation for virtually all of modern mathematics, and as such it is helpful to have at least an intuitive grasp of it. Set theory and logic supplementary materials math 103. Model theory discusses the properties such classes of interpretations have. Hc of an admissible set, journal of symbolic logic, vol. Set theory and logic dover books on mathematics reprint. 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. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning graduate level. After mathematical logic has provided us with the methods of reasoning we start with a very basic theory. The period between the two world wars is the beginning of the classical time of logic which at some point in the second half of the 20th century turned into baroque. Computational logic and set theory applying formalized. They are not guaranteed to be comprehensive of the material covered in the course.
However, most of the time, we only have an intuitive picture of what set theory should look like there are sets, we can take intersections, unions, intersections and subsets. 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. Studies in logic and the foundations of mathematics. 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. We will return to sets as an object of study in chapters 4 and 5. There are many other details and plenty of terminology that i. Unit 1 set theory and logic miss douglas class website. If the object x is a member of the set a, then we write x a which is read as x is a member of aorx belongstoaorx is in aorx is an. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Logic and set theory wikibooks, open books for an open world. The axiom of pair, the axiom of union, and the axiom of.
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