Mathematical logic and set theory math 220a, fall quarter 2016 m 22. 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. 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. In mathematical logic, a theory also called a formal theory is a set of sentences in a formal language that is closed under logical implication. The system we pick for the representation of proofs is gentzens natural deduction, from 8. Priors formal logic, start things from logic without any set theory required. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Math 522, advanced set theory, at boise state university, on the. This dictates that we develop logic itself formally, and subsequently set theory as a formal theory. Mathematical logic as based on the theory of types. If we seek help by deciding that no class is a member of itself, then w becomes the class of. 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.
While the user of mathematical logic will practise the various proof. Set theory is indivisible from logic where computer science has its roots. Complex issues arise in set theory more than any other area of pure mathematics. Determine if certain combinations of propositions are. In this introductory chapter we deal with the basics of formalizing such proofs. A first course in mathematical logic and set theory. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. In mathematics, the notion of a set is a primitive notion. The topics in this course are part of the common background of mathematicians active in any one of these areas. The relation on the set of integers is reflexive and transitive.
Development of the fundamental axiom systems for mathematics 1880s1920s. Can set theory for example, currently, the most popular version of it zermelofraenkels set theory be viewed as a common logic for the classical traditional mathematics. Mathematical logic is a group of interrelated mathematical subjects which. The exciting and active areas of logic today are set theory, model theory and connections with computer science. Logic the main subject of mathematical logic is mathematical proof. Further, i have become more interested in category theory as of late and was hoping someone could refer to me a good introduction on the subject. I have been on the prowl for a concise introduction to the above stated fields of mathematics.
An introduction to set theory university of toronto. For example, a deck of cards, every student enrolled in. Logic and set theory virginia commonwealth university. Basic concepts the concept of set is considered a primitive concept in math a set is a collection of elements whose description must be unambiguous and unique. Mathematical logic and set theory ucla department of. Discrete mathematics, set theory, and logic discrete math, graph theory, combinatorics, recurrence relations difference equations, set theory, gene. You may think so, if you do not wish to distinguish between the first order notion of natural numbers. Some familiarity with logic and set theory is required for studying calculus also, but as far as a systematic study of these topics are concerned, you should do it if you are really interested in it. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. For those of you new to abstract mathematics elementary does not mean simple though much of the material. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. A first course in mathematical logic and set theory wiley.
In most scenarios, a deductive system is first understood from context, after which an element. So it is not possible to separate the history of set theory and the formalisation of mathematics from that of mathematical logic. Exercise classes where students solve tasks and problems from logic and set theory form of assessment. I am not too concerned with most other fields of math, rather only the foundational stuff. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Although any type of object can be collected into a set. From the o cial course description of the department. For any xand y, x,y is the set whose members are exactly xand y. These notes for a graduate course in set theory are on their way to be coming a book. Model theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics. Lukasiewiczs book takes you up to firstorder predicate logic, while priors book takes you up to set theory, though i havent read that far personally in the book, and will. An elementary introduction to logic and set theory. Set theory chiara ghidini fbkirst, trento, italy 20142015 chiara ghidini mathematical logic outline set theory relations functions 1 set theory basic concepts operations on sets operation properties 2 relations properties equivalence relation 3 functions properties chiara ghidini mathematical logic outline. Meant as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy, this twovolume work is written in a userfriendly conversational lecture style that makes it equally effective for selfstudy or class use.
The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with. Mathematical theories are expressed using first order logic. Set theory relations functions basic concepts operations on sets operation properties sets. You wont be needing very advanced knowledge of logic for studying topics like. 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. Volume ii, on formal zfc set theory, incorporates a selfcontained chapter 0 on proof techniques based on formal logic, in the. Textbook examples will serve as solution models to most of the exercise questions at. The formal character of mathematical statements and arguments is the basis for the selfmodelling of mathematics in mathematical logic. One feature of the proof theory is that we deal with both common approaches to the treatment of nonsentence formulae, giving the appropriate deduction. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Highlighting the applications and notations of basic mathematical concepts within the framework of logic and set theory, a first course in mathematical logic and set theory introduces how logic is used to prepare and structure proofs and solve more complex problems.
Other area of pure mathematics in particular, mathematical logic is used in a fundamental. Elements of logic and set theory university of bialystok. Mathematical logic is a branch of mathematics which is mainly concerned with the relationship between semantic. No student is expected to really benefit from these notes unless they have regularly attended the lectures. Outline set theory relations functions 1 set theory basic concepts operations on sets operation properties.
Mathematical logic is a branch of mathematics which is mainly concerned with the relationship between semantic concepts i. Mathematical logic applies to mathematical reasoning the art and science of writing down deductions. Curiously, set theory arose in the context of sets of real numbers related to the convergence of fourier series. You wont be needing very advanced knowledge of logic for studying topics like abstract algebra, linear algebra, analysis etc.
Set theory chiara ghidini fbkirst, trento, italy 202014 chiara ghidini mathematical logic. Nowadays, mathematical logic is often divided into the four elds of set theory, model theory, recursion theory or computability theory and proof theory. Basic set theory a set is a many that allows itself to be thought of as a one. I have found logic to be quite interesting, however i am looking for a good book on the subject alongside basic set. 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. Textbook for students in mathematical logic and foundations of mathematics. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. For any set x, there is a set to which all members of members of xbelong.
A mathematical introduction to the theory and applications of logic and set theory. A mathematical introduction to the theory and applications of logic and set theory with an emphasis on writing proofs. We give below examples for three types of fallacies that are often found in attempted mathematical proofs. The axioms of pairing, union, and comprehension give us some operations on sets. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. They are not guaranteed to be comprehensive of the material covered in the course. It is evident that we need a precise formulation of set theory, that is, we must turn it into a mathematical object in order to make task 2, above, a meaningful mathematical activity.
To learn how to formally construct and model mathematical objects on settheoretical grounds. Set theory addresses various ways to axiomatize mathematics, with implications for understanding the properties of sets having large infinite cardinalities and connections with the axiomatization. B for example, if s is the set of all juices in the supermarket, and t is the set of all. Teaching methods and the overall form of credit for the course lectures. Set theory and logic supplementary materials math 103. Propositional logic propositional logic is a mathematical system for reasoning about propositions and how they relate to one another. Set theory is likely to be around long after most presentday programming languages have faded from memory. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student.
But traditional logic, like that of the modern philosophers, covers in principle, an area of applications far greater than mathematics. In many deductive systems there is usually a subset. William weiss and cherie dmello university of toronto. Proof, sets, and logic department of mathematics boise state. To discuss examples it is convenient to introduce some notation. 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. Propositional logic enables us to formally encode how the truth of various propositions influences the truth of other propositions.
Contact us 100 math tower 231 west 18th avenue columbus, oh 432101174. Set theory for computer science university of cambridge. Introduction to logic and set theory 202014 bgu math. The fundamental tenet of model theory is that mathematical truth, like all truth, is relative. I have found logic to be quite interesting, however i am looking for a good book on the subject alongside basic set theory. This volume is about the form, meaning, use, and limitations of logical deductions, also called proofs.
If we seek help by deciding that no class is a member of itself, then w becomes the class of all classes, and we have to decide. Numerous examples that illustrate theorems and employ basic concepts such as euclids lemma, the fibonacci sequence, and unique factorization. To learn how to formally construct and model mathematical objects on set theoretical grounds. Axioms and set theory mathematics university of waterloo. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools.
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