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


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