f0;2;4;:::g= fxjxis an even natural numbergbecause two ways of writing a set are equivalent. These notes for a graduate course in set theory are on their way to be-coming a book. We refer to  for a historical overview of the logic and the set theory developments at that time given in the form of comics. IV. 4. 3. For our purposes, it will sufce to approach basic logical concepts informally. In mathematics, the notion of a set is a primitive notion. There is a natural relationship between sets and logic. A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. They are not guaran-teed to be comprehensive of the material covered in the course. Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. Similarly, we want to put logic 1. axiomatic set theory with urelements. These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. Cynthia Church pro-duced the ﬁrst electronic copy in December 2002. Predicates. x2Adenotes xis an element of A. N = f0;1;2;:::gare the natural numbers. Each of the axioms included in this the- They originated as handwritten notes in a course at the University of Toronto given by Prof. William Weiss. Many of the elegant proofs and exam- If A is a set, then P(x) = " x ∈ A '' is a formula. 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 Negation of Quantified Predicates. 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. Tautologies. Predicate Logic and Quantifiers. Indirect Proof. Multiple Quantifiers. Closely related to set theory is formal logic. The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to define what a set is, … Propositions. IV. Conditional. Conjunction. 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. Universal and Existential Quantifiers. set theory. Universal and Existential Quantifiers. … The study of these topics is, in itself, a formidable task. 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. 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. Predicate Logic and Quantifiers. Chapters 1 to 9 are close to ﬁ- Sentential Logic. Negation. P. T. Johnstone, ‘Notes on Logic & Set Theory’, CUP 1987 2. f1;2;3g= f3;2;2;1;3gbecause a set is not de ned by order or multiplicity. ;is the empty set. Mathematical Induction. Also, their activity led to the view that logic + set theory can serve as a basis for 1 I. Overview. Sets and elements Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. D. Van Dalen, ‘Logic and Structure’, Springer-Verlag 1980 (good for Chapter 4) 3. The standard form of axiomatic set theory is the Zermelo-Fraenkel set theory, together with the axiom of choice. A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Q = fm n The subjects of register machines and random access machines have been dropped from Section 5.5 Chapter 5. Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. Disjunction. Basic Concepts of Set Theory. Informal Proof. 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. Methods of Proof. 5. 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. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the … Proof by Counter Example. II. V. Naïve Set Theory. both the logic and the set theory on a solid basis. 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… Methods of Proof The language of set theory can be used to define nearly all mathematical objects. Z = f:::; 2; 1;0;1;2;:::gare the integers. Predicates. An Elementary Introduction to Logic and Set Theory. That is, we adopt a naive point of view regarding set theory and assume that the meaning of 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. 1.1. 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. James Talmage Adams produced the copy here in February 2005. III. Primitive Concepts. This is similar to Euclid’s axioms of geometry, and, in some sense, the group axioms. Negation of Quantified Predicates. Unique Existence. III. Set Theory is indivisible from Logic where Computer Science has its roots. Conditional Proof. 1 Elementary Set Theory Notation: fgenclose a set. Biconditional. Multiple Quantifiers. Formal Proof. Unique Existence. A. Hajnal & P. Hamburger, ‘Set Theory’, CUP 1999 (for cardinals and ordinals) 4.
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