Fundamentals of mathematical logic logic is commonly known as the science of reasoning. Some of the reasons to study logic are the following. 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. As in the above example, we omit parentheses when this can be done without ambiguity. Our objective is to reduce the process of mathematical reasoning, i. Lecture notes on mathematical logic vladimir lifschitz january 16, 2009 these notes provide an elementary, but mathematically solid, introduction to propositional and. Nevertheless, we will try our best not to lose our basic intuition by making occasional remarks to the. The most useful class of tautologies are logical equivalences. Excellent as a course text, the book presupposes only elementary background and can be used also for selfstudy by more ambitious students. In mathematics, the notion of a set is a primitive notion.
Each variable represents some proposition, such as you wanted it or you should have put a ring on it. I have no con dence in these, but one must ask something. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. 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. At the hardware level the design of logic circuits to implement in. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with. In this text we study mathematical logic as the language and deductive system of mathematics and computer science. Validity, entailment, and equivalence of propositions revisited. We start with a brief overview of mathematical logic as covered in this course. The intended purpose is to cover the basics of modal logic from a rather mathematical perspective.
Examples of structures the language of first order logic is interpreted in mathematical structures, like the following. This might include proving things about proving things. 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. Lecture notes for math2040 mathematical logic 1 michael rathjen school of mathematics university of leeds autumn 2009 lecture notes for math2040 mathematical logic 1. Note that the expansion of the language by these congruence relations is. Elements of a set can be just about anything from real physical objects to abstract mathematical objects. There are now exercise sets at the ends of sections 3. The exercise solutions have not been carefully checked. Introduction to logic and set theory 202014 bgu math. Pdf discrete structures notes lecture free download. The central concept of deductive logic is the concept of argument form. Popular lectures on mathematical logic dover books on. The system we pick for the representation of proofs is gentzens natural deduction, from 8.
Lecture notes department of theoretical computer science. Complex issues arise in set theory more than any other area of pure mathematics. Introduction maybe not all areas of human endeavour, but certainly the sciences presuppose an underlying acceptance of basic principles of logic. Mathematical logic introduction mathematics is an exact science. They are not guaranteed to be comprehensive of the material covered in the course. Lecture notes on mathematical logic computer science. Mathematical logic for computer science is a mathematics textbook, just as a. A key result known as the compactness theorem5 states that a set s of lsentences is satis. The course unit handles concepts such as logic, methods of proof, sets, functions, real number properties, sequences and series, limits and continuity. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Basic set theory members of the collection comprising the set are also referred to as elements of the set.
An argument is a sequence of statements aimed at demonstrating the truth of an assertion a claim. The following related notion is of technical importance. Removed the rst approach to collapsing names and cleaned up. An introduction to set theory university of toronto. The lecture notes section contains 22 lecture slides, 37 inclass problems, 37 solutions to inclass problems, and 2 supplements for the course. This is a systematic and wellpaced introduction to mathematical logic. An introduction to writing proofs, the basic types of proofs, and an introduction to important mathematical objects such as functions and relations.
They may not have much in common in the way of subject matter or methodology but what they have in common. Logic for the mathematical course notes for pmath 330spring2006 peter hoffman peter ho. Math 557 is an introductory graduatelevel course in mathematical logic. Removed the rst approach to collapsing names and cleaned up typos and editing disasters found during lecture.
We will develop some of the symbolic techniques required for computer logic. Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. The logic and set theory are presented in a naive way. Propositional logic is a formal mathematical system whose syntax is rigidly specified.
Preface this book is designed for a one semester course in discrete mathematics for sophomore or junior level students. The emphasis here will be on logic as a working tool. Basic concepts of set theory, functions and relations. Lecture notes logic i linguistics and philosophy mit. Mathematical logic is the study of mathematical reasoning.
Part is devoted to the detailed construction of our \model of reasoning for rstorder languages. In logical metatheory, rather using a logical system to construct a proof about. They are not guaran teed to be comprehensive of the material covered in the course. 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. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. Logic the main subject of mathematical logic is mathematical proof. In this introductory chapter we deal with the basics of formalizing such proofs.
These are lecture notes for ame 60611 mathematical methods i, the. These notes provide an elementary, but mathematically solid, introduc. Hence, there has to be proper reasoning in every mathematical proof. Every statement in propositional logic consists of propositional variables combined via logical connectives. It is part of the metalanguage rather than the language. The deduction system consisting of the logical axiom schemes above is sound and complete. The basic set operations union, intersection and complement on subsets of a xed set.
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. These lecture notes were prepared for the turkish mathematical society 2009 summer school. The study of logic helps in increasing ones ability of systematic and logical reasoning. Wangs presentations were wellreceived and subsequently published in 1981 under the title popular lectures on mathematical logic. Textbook for the lecture automata and grammars czech. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Contribute to purtroppomathematical logic development by creating an account on github. Kueker university of maryland, college park email address. It also helps to develop the skills of understanding various statements and their validity. In these notes we will study rstorder languages almost exclusively. This is a set of lecture notes for introductory courses in mathematical logic offered at the pennsylvania state.
It is one of two firstyear graduate courses in mathematical logic, the other being math 558. Most of the students in this course are beginning graduate students in engineering coming from a variety of backgrounds. It is suitable for all mathematics graduate students. Find materials for this course in the pages linked along the left. The british mathematician and philosopher george boole 18151864 is the man who made logic mathematical. The study of logic helps in increasing ones ability of. Simpson, a mathematician at penn state university lecture notes. Logic has a wide scale application in circuit designing, computer programming etc. His book the mathematical analysis of logic was published in 1847. Starting with the basics of set theory, induction and computability, it covers. Lecture notes for math 2040 mathematical logic i semester 1, 200910 michael rathjen chapter 0.
Lecture notes mathematics for computer science electrical. The lecture notes section contains 22 lecture slides, 37 inclass problems, 37 solutions to in. 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. A 1993 reprint edition by dover includes a useful postscript in which wang briefly outlines recent advances in mathematical logic. These notes provide an elementary, but mathematically solid, introduction to propositional and.
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