… Predicates. For our purposes, it will sufce to approach basic logical concepts informally. These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. Methods of Proof. This is similar to Euclid’s axioms of geometry, and, in some sense, the group axioms. Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Conditional Proof. Mathematical Induction. There is a natural relationship between sets and logic. Indirect Proof. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Conditional. Unfortunately, while axiomatic set theory appears to avoid paradoxes like Russel’s paradox, as G odel proved in his incompleteness theorem, we cannot prove that our axioms are free of contradictions. Primitive Concepts. 1 Elementary Set Theory Notation: fgenclose a set. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Propositions. If A is a set, then P(x) = " x ∈ A '' is a formula. Many of the elegant proofs and exam- Chapters 1 to 9 are close to ﬁ- set theory. Closely related to set theory is formal logic. ;is the empty set. Negation. Basic Concepts of Set Theory. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Formal Proof. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the … Universal and Existential Quantifiers. Q = fm n The subjects of register machines and random access machines have been dropped from Section 5.5 Chapter 5. Negation of Quantified Predicates. Predicate Logic and Quantifiers. The study of these topics is, in itself, a formidable task. The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to define what a set is, … Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. III. Universal and Existential Quantifiers. 5. f0;2;4;:::g= fxjxis an even natural numbergbecause two ways of writing a set are equivalent. Methods of Proof 1.1. P. T. Johnstone, ‘Notes on Logic & Set Theory’, CUP 1987 2. V. Naïve Set Theory. Proof by Counter Example. Unique Existence. 1. In mathematics, the notion of a set is a primitive notion. We refer to [1] for a historical overview of the logic and the set theory developments at that time given in the form of comics. Also, their activity led to the view that logic + set theory can serve as a basis for 1 That is, we admit, as a starting point, the existence of certain objects (which we call sets), which we won’t deﬁne, but which we assume satisfy some II. Sets and elements Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. both the logic and the set theory on a solid basis. That is, we adopt a naive point of view regarding set theory and assume that the meaning of The language of set theory can be used to define nearly all mathematical objects. The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to definewhat a set is, but we can give an informal description, describe important properties of sets… Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. It has been and is likely to continue to be a a source of fundamental ideas in Computer Science from theory to practice; Computer Science, being a science of the articial, has had many of its constructs and ideas inspired by Set Theory. Z = f:::; 2; 1;0;1;2;:::gare the integers. Predicates. f1;2;3g= f3;2;2;1;3gbecause a set is not de ned by order or multiplicity. One can mention, for example, the introduction of quanti ers by Gottlob Frege (1848-1925) in 1879, or the work By Bertrand Russell (1872-1970) in the early twentieth century. Conjunction. An Elementary Introduction to Logic and Set Theory. 4. They are not guaran-teed to be comprehensive of the material covered in the course. Predicate Logic and Quantifiers. Similarly, we want to put logic Informal Proof. III. A. Hajnal & P. Hamburger, ‘Set Theory’, CUP 1999 (for cardinals and ordinals) 4. Multiple Quantifiers. An appendix on second-order logic will give the reader an idea of the advantages and limitations of the systems of first-order logic used in Chapters 2-4, and will provide an introduction to an area of much current interest. Disjunction. A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. These notes for a graduate course in set theory are on their way to be-coming a book. Each of the axioms included in this the- It is true for elements of A and false for elements outside of A. Conversely, if we are given a formula Q(x), we can form the truth set consisting of all x that make Q(x) true. Tautologies. x2Adenotes xis an element of A. N = f0;1;2;:::gare the natural numbers. IV. I. Overview. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. Cynthia Church pro-duced the ﬁrst electronic copy in December 2002. A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Set Theory is indivisible from Logic where Computer Science has its roots. IV. Sentential Logic. They originated as handwritten notes in a course at the University of Toronto given by Prof. William Weiss. The standard form of axiomatic set theory is the Zermelo-Fraenkel set theory, together with the axiom of choice. axiomatic set theory with urelements. Negation of Quantified Predicates. Biconditional. Unique Existence. James Talmage Adams produced the copy here in February 2005. From our perspective we see their work as leading to boolean algebra, set theory, propositional logic, predicate logic, as clarifying the foundations of the natural and real number systems, and as introducing suggestive symbolic notation for logical operations. Multiple Quantifiers. D. Van Dalen, ‘Logic and Structure’, Springer-Verlag 1980 (good for Chapter 4) 3. 3.