I have no con dence in these, but one must ask something. It also helps to develop the skills of understanding various statements and their validity. Logic the main subject of mathematical logic is mathematical proof. Introduction maybe not all areas of human endeavour, but certainly the sciences presuppose an underlying acceptance of basic principles of logic. These lecture notes were prepared for the turkish mathematical society 2009 summer school. A 1993 reprint edition by dover includes a useful postscript in which wang briefly outlines recent advances in mathematical logic. Starting with the basics of set theory, induction and computability, it covers. Contribute to purtroppomathematical logic development by creating an account on github. Textbook for the lecture automata and grammars czech. In these notes we will study rstorder languages almost exclusively. The study of logic helps in increasing ones ability of. Mathematical logic introduction mathematics is an exact science.
They are not guaran teed to be comprehensive of the material covered in the course. Every statement in propositional logic consists of propositional variables combined via logical connectives. The following related notion is of technical importance. Introduction to logic and set theory 202014 bgu math. Basic set theory members of the collection comprising the set are also referred to as elements of the set. Lecture notes mathematics for computer science electrical. His book the mathematical analysis of logic was published in 1847. In logical metatheory, rather using a logical system to construct a proof about. It is suitable for all mathematics graduate students. The study of logic helps in increasing ones ability of systematic and logical reasoning. Lecture notes for math 2040 mathematical logic i semester 1, 200910 michael rathjen chapter 0. Removed the rst approach to collapsing names and cleaned up.
Some of the reasons to study logic are the following. These notes were prepared as an aid to the student. Firstorder languages are the most widely studied in modern mathematical logic, largely to obtain the bene t of the completeness theorem and its applications. Excellent as a course text, the book presupposes only elementary background and can be used also for selfstudy by more ambitious students. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. This is a set of lecture notes for introductory courses in mathematical logic offered at the pennsylvania state. Find materials for this course in the pages linked along the left. The emphasis here will be on logic as a working tool.
We will develop some of the symbolic techniques required for computer logic. These are lecture notes for ame 60611 mathematical methods i, the. Lecture notes on mathematical logic vladimir lifschitz january 16, 2009 these notes provide an elementary, but mathematically solid, introduction to propositional and. 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 deduction system consisting of the logical axiom schemes above is sound and complete. The lecture notes section contains 22 lecture slides, 37 inclass problems, 37 solutions to in. Wangs presentations were wellreceived and subsequently published in 1981 under the title popular lectures on mathematical logic. This is a systematic and wellpaced introduction to mathematical logic. The basic set operations union, intersection and complement on subsets of a xed set. Lecture notes logic i linguistics and philosophy mit. These notes provide an elementary, but mathematically solid, introduction to propositional and. In mathematics, the notion of a set is a primitive notion.
The logic and set theory are presented in a naive way. Removed the rst approach to collapsing names and cleaned up typos and editing disasters found during lecture. In logical metatheory, rather using a logical system to construct a proof about another subject matter or an unknown one, we prove things about logic itself. Fundamentals of mathematical logic logic is commonly known as the science of reasoning. Popular lectures on mathematical logic dover books on. Logic has a wide scale application in circuit designing, computer programming etc. Hence, there has to be proper reasoning in every mathematical proof. Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. Logic for the mathematical course notes for pmath 330spring2006 peter hoffman peter ho.
The lecture notes section contains 22 lecture slides, 37 inclass problems, 37 solutions to inclass problems, and 2 supplements for the course. Basic concepts of set theory, functions and relations. This might include proving things about proving things. Our objective is to reduce the process of mathematical reasoning, i. Simpson, a mathematician at penn state university lecture notes.
The system we pick for the representation of proofs is gentzens natural deduction, from 8. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. Complex issues arise in set theory more than any other area of pure mathematics. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. The british mathematician and philosopher george boole 18151864 is the man who made logic mathematical. The exercise solutions have not been carefully checked. In these discrete structures notes pdf, you will study the fundamental concepts of sets, relations and functions, mathematical logic, group theory, counting theory, probability, mathematical induction and recurrence relations, graph theory, trees and boolean algebra. Lecture notes on mathematical logic computer science.
They may not have much in common in the way of subject matter or methodology but what they have in common. Math 557 is an introductory graduatelevel course in mathematical logic. Part is devoted to the detailed construction of our \model of reasoning for rstorder languages. Large cardinals, determinacy and other topics is the final volume in a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of research developments since the.
Examples of structures the language of first order logic is interpreted in mathematical structures, like the following. There are now exercise sets at the ends of sections 3. The intended purpose is to cover the basics of modal logic from a rather mathematical perspective. They are not guaranteed to be comprehensive of the material covered in the course. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with. Kueker university of maryland, college park email address. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Each variable represents some proposition, such as you wanted it or you should have put a ring on it. Mathematical logic is the study of mathematical reasoning. An introduction to writing proofs, the basic types of proofs, and an introduction to important mathematical objects such as functions and relations. A key result known as the compactness theorem5 states that a set s of lsentences is satis. The most useful class of tautologies are logical equivalences. Lecture notes department of theoretical computer science.
Nevertheless, we will try our best not to lose our basic intuition by making occasional remarks to the. Note that the expansion of the language by these congruence relations is. Pdf discrete structures notes lecture free download. An argument is a sequence of statements aimed at demonstrating the truth of an assertion a claim. The central concept of deductive logic is the concept of argument form. In this introductory chapter we deal with the basics of formalizing such proofs.
We start with a brief overview of mathematical logic as covered in this course. In this text we study mathematical logic as the language and deductive system of mathematics and computer science. An introduction to set theory university of toronto. Mathematical logic for computer science is a mathematics textbook, just as a. Elements of a set can be just about anything from real physical objects to abstract mathematical objects. Validity, entailment, and equivalence of propositions revisited. These notes provide an elementary, but mathematically solid, introduc. It was introduced by moses schonfinkel1 and haskell curry, 2 and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages.
As in the above example, we omit parentheses when this can be done without ambiguity. Preface this book is designed for a one semester course in discrete mathematics for sophomore or junior level students. Propositional logic is a formal mathematical system whose syntax is rigidly specified. It is part of the metalanguage rather than the language. It is one of two firstyear graduate courses in mathematical logic, the other being math 558. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin.
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