In this chapter we discuss the need for a language more formal than common language to write proofs. In an attempt to avoid these paradoxes, set theory was axiomatized based on firstorder logic, and thus axiomatic set theory was born. Like logic, the subject of sets is rich and interesting for its own sake. We proceed to introduce propositional logic, quantifiers, and the basics of the language of set theory, including functions, onetoone and onto functions, and their use in. An explanation of the branch of logic known as set theory which deals with groups of objects and serves as the foundations of mathematics. In mathematics, the notion of a set is a primitive notion. We will return to sets as an object of study in chapters 4 and 5.
Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Discrete math, graph theory, combinatorics, recurrence relations difference equations, set theory, general and symbolic logic. If f and p are statements that depend on a variable x, then we define the sets x f x is true and x p x is true. They are not guaranteed to be comprehensive of the material covered in the course. We observe the following formal correspondence between logic and set theory.
A set up for the cathegorical langlands correspondence. Nylogic set theory seminar model theory seminar logic workshop mopa mamls. Finally, there are questions about which logic should be used to capture our conception of the set theoretic universe and why. Everyday low prices and free delivery on eligible orders. Sets much like the everyday use of the term are collections of objects, which is handy. Defining logic is a bit challenging and it is more like a philosophical endeavor but concisely speaking it is a system rules inference rules that can help us prove and disprove stuff. Choi kaist logic and set theory november 20, 2012 4 14. This 1963 book by robert roth stoll is more than twice as big as the authors 1961 sets, logic and axiomatic theories. Sets are usually described using fg and inside these curly brackets a list of the elements or a description of the elements of the set. No student is expected to really benefit from these notes unless they have regularly attended the lectures. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin.
In higher order logic, the quantifiers may refer to collections of objects, or to collections of formulas about objects. Logic and set theory virginia commonwealth university. All instructor resources see exceptions are now available on our instructor hub. In first order logic, all quantifiers for all and there exists refer always to objects in the theory elements in group theory, sets in set theory, etc. Nebent introduction to set theory and logic literaturverz. If ais an element of a set a, we use the notation a2aand often say ain a instead of aan element of a. Mathematical logic is a branch of mathematics which is mainly concerned with the relationship between semantic concepts i.
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. For those of you new to abstract mathematics elementary does not mean simple though much of the material. The logic can be confused by rationality or pragmatism, whil. Part ii logic and set theory based on lectures by i. Set theory and logic textbook binding january 1, 2000 by a. 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. Well, theres a sort of sense in which theyre equivalent. Set theory relations functions basic concepts operations on sets operation properties sets. For example, a deck of cards, every student enrolled in math 103, the collection of all even integers, these are all examples of sets of things. Almost everyone knows the game of tictactoe, in which players mark xs and os on a threebythree grid until one player makes three in a row, or the grid gets filled up with no winner a draw.
The omitting types theorem of infinitary logic is used to prove that every small. Students need to be familiar with the following concepts. 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. But set theory as you might get from the name deals with the structure and operations of sets. Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. The standard book is paul halmos, naive set theory 1960. Remember, no credit is given for any marks on the answer sheet except the. The garland science website is no longer available to access and you have been automatically redirected to.
In pure mathematics, it is the place where questions about in. This is an ideal painless introduction to standard logic and set theory for anyone with a couple of years of undergraduate pure mathematics background. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. In this video we give a very quick overview of a highly controversial period in the development of modern mathematics. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Ultimately, the goal of set theory was to provide a common axiomatic basis for all of mathematics.
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. It will cover truth tables, logical equivalences, validity and invalidity, soundness, sets, set operations, venn. Curiously, set theory arose in the context of sets of real numbers related to the convergence of fourier series. Logic and set theory a rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. For most purposes, however, naive set theory is still useful. This is a very, very brief overview, for more detail id recommend the stanford encyclopaedia page and for a book length treatment m. Although this is a fascinating study of permanent interest, it does not account for the importance of set theory in applied areas. In this lecture, we will be more rigorous than in htp and use axioms to establish facts. Fraenkel author see all 2 formats and editions hide other formats and editions. 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. In this introduction to set theory and logic, the author discusses first order logic, and gives a rigorous axiomatic presentation of zermelofraenkel set theory.
The whiteheadrussell way of doing logic and set theory has long been relegated to the history books, while zf set theory and its spawn have dominated the scene. 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. One of the most complex and essential of modern mathematical. He includes many methodological remarks and explanations, and demonstrates how the basic concepts of mathematics can be reduced to set theory. The logical connective that represents this operator is typically written as. My solutions to stolls set theory and logic textbook. Propositional logic, interpretation with respect to a model expressed in terms of sets. Discrete mathematics, set theory, and logic math help boards. The introduction to nlp module assumes some basic knowledge of set theory and logic. Concentration music with binaural beats, focus music for studying, study music greenred productions relaxing music 1,378 watching live now.
Set theory was developed in the second half of the nineteenth century. If the correct answer is not one of the choices, mark e on teh answer sheet. Lectures in logic and set theory library of congress. 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.
Select the letter of the most appropriate answer and shade in the corresponding region of the answer sheet. What is the difference between set theory and logic. There the importance stems from the fact that set theory provides an incredibly versatile. An overview of logic, proofs, set theory, and functions. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Basic set theory lx 502 semantics i september 11, 2008 1. Quine, set theory, consistent subsystems, bibliography, universal set notes on logic, proofs, and set theory at csusb add. The book is based on lectures given at the university of cambridge and covers the basic concepts of logic. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. This hardly needs saying, since if it is mathematicians notion of theory, then it is a mathematical and therefore logical consequence of the fact that a logician is a mathematician. 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.
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. Unlike axiomatic set theories, which are defined using a formal logic, naive set theory is defined informally, in natural language. 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 this module weve seen how logic and valid arguments can be formalized using mathematical notation and a few basic rules. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics.
Logic and set theory year 2019 2018 2017 2016 2015 2014 20 2012 2011 2010 2009 2008 2007 2006 2005. Plot functions, solve a diophantine equation, modulo arithmetic practice, truth table practice, symbolic logic a problem course in mathematical logic add. You wont be needing very advanced knowledge of logic for studying topics like abstract algebra, linear algebra, analysis etc. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. July 3 the seminar will take place virtually at 2pm us eastern standard time.
Borrowing money chapter 3 set theory and logic chapter 4 counting methods chapter 5 probability chapter 6 polynomial functions chapter 7 exponential and logarithmic functions. Only informal, or nave, set theory notation and results are needed in chapter i at the metalevel, i. The objects in a set will be called elements of the set. Set theory \a set is a many that allows itself to be thought of as a one. Set theory and logic dover books on mathematics reprint. It has its roots in the work of georg cantor, although contributions of others such as gottlob frege and giuseppe peano were significant. What is the difference between algebra, logic and set theory. Basic concepts of set theory, functions and relations. The old theory of types is still discussed somewhat in the philosophy department version of mathematical logic, but the mathematics department principally remembers russells work only. B for example, if s is the set of all juices in the supermarket, and t is the set of all.
Department of pure mathematics and mathematical statistics. Some further remarks about set theory references symbols and notation author index subject index 373 373 375 387 394 401 409 417 426 436 446 452 457 465 467 469. 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. And a theory, in firstorder logic anyway, is a deductively closed set of sentences.
Which axioms of set theory are needed to prove this sentence. Though propositional logic will prove a useful tool to describe certain aspects of meaning, like the reasoning in 1, it is a blunt. An elementary introduction to logic and set theory. Sentential logic we go into the mathematical theory of the simplest logical notions. 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. It will be suitable for all mathematics undergraduates coming to the subject for the first time. Perhaps landau hoped to induce the student to read the preface for the teacher, but not to worry about digesting. The study of these topics is, in itself, a formidable task.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Set theory and logic thomas forster this course is the sequel to the part ii course with the title logic and set theory. A taste of set theory for philosophers helsingin yliopisto. Some people consider naive set theory to be what the 19th century had to offer. The language of set theory can be used to define nearly all mathematical objects. Basic set theory a set is a many that allows itself to be thought of as a one.