The discussion is su pported by a wide range of exercises. Set theory and logic is the result of a course of lectures for advanced undergraduates, developed at oberlin college for the purpose of introducing students to the conceptual foundations of mathematics. Consistency, completeness, and categoricity of firstorder theories 8. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is, but we can give an informal description, describe important properties of sets, and give. Set theory and logic dover books on mathematics reprint. Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the axiom of choice, godels theorem, and the skolem. Further, by focusing attention on certain chapters, it can serve as a text for one semester courses in set. Introduction to logic and set theory 202014 bgu math. After mathematical logic has provided us with the methods of reasoning we start with a very basic theory. We will return to sets as an object of study in chapters 4 and 5. Thus, in an axiomatic theory of sets, set and the membership relation.
Inference within the science of logic 205 inference within axiomatic systems. Though aimed at a nonmathematical audience in particular, students of philosophy and computer science, it is rigorous. We shall also discuss cantors continuum problem which is the following. In this section we discuss axiomatic systems in mathematics.
Also first order logic is developed in the metatheory. A philosophical perspective on logic as a whole 218 the indispensability of modal concepts within propositional logics 218. One of our main aims in this course is to prove the following. Sets, logic and axiomatic theories by robert roth stoll. Christoph benzmuller, dale miller, in handbook of the history of logic, 2014.
As we have seen from our examples, sets may contain a. Use features like bookmarks, note taking and highlighting while reading set theory and logic dover books on mathematics. 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 notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is, but we can give an informal description, describe. Consistency and satisfiability of sets of formulas 7. Pdf the interpretation of classes in axiomatic set theory. Studies in logic and the foundations of mathematics. Axiomatic set theorems are the axioms together with statements that can be deduced from the axioms using the rules of inference provided by a system of logic. We explain the notions of primitive concepts and axioms. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. Proof, sets, and logic department of mathematics boise state. Buy sets logic and axiomatic theories edition undergraduate mathematics books on.
The assumptions adopted about these notions are called the axioms of the theory. They are not guaranteed to be comprehensive of the material covered in the course. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. The axioms of set theory of my title are the axioms of zermelofraenkel set theory, usually thought ofas arisingfromthe endeavourtoaxiomatise the cumulative hierarchy concept of set. Cantini andrea, in handbook of the history of logic, 2009. The elementary part of symbolic logic, the statement calculus, is. Set theory for computer science university of cambridge. From a wholly different perspective, axiomatic theories of selfreferential truth, which formalize the closure conditions related to kripkes jump operator, were investigated by feferman in connection with an unramified approach to predicativity and with the general problem of the incompleteness of formal. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Alternative axiomatic set theories stanford encyclopedia of.
We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next. Set theory is first treated naively an axiomatic treatment is given after the basics of firstorder logic have been introduced. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. The question of infinity had long been one of the most sensitive problems of mathematics. Axioms and set theory mathematics university of waterloo. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory, logic, symbolic and mathematical publisher san francisco, w. The problem with the former two theories is that they seem to require at least secondorder quanti cation, e. Working in an axiomatic set theory such as zfc, one can formalize the syntax of logic and then prove theorems about model theory. About the open logic project the open logic text is an opensource, collaborative textbook of formal metalogic and formal methods, starting at an intermediate level i. Then a bif they contain exactly the same elements, that is a2a a2b. Like logic, the subject of sets is rich and interesting for its own sake. In mathematics, a set is a welldefined collection of distinct objects, considered as an object in its own right. Given that fewer axioms are needed in stt than in axiomatic set theory, and that the term and formula structure is enriched, some earlier researchers in automated theorem proving were attracted to stt since traditionally such early provers were not well suited to deal with.
Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. The relationship of admissible sets to logic can be. Axioms of set theory and logic mathematics stack exchange. In mathematics, the notion of a set is a primitive notion. To prove that two sets aand bare equal, we need to show that for all a2awe have a2band for all a2b, we have a2a. However, most of the time, we only have an intuitive picture of what set theory should look like there are sets, we can take intersections, unions, intersections and subsets.
Basic concepts of set theory, functions and relations. Axiomatic theories of truth stanford encyclopedia of philosophy. Intuitive set theory is treated in detail with numerous examples and exercises. Rr stoll, sets, logic, and axiomatic theories cambridge. Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians.
His aim was to save axiomatic set theory by overcoming some of its defects. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written 2, 4, 6. Publication date 1974 topics logic, symbolic and mathematical, set theory. Although the necessary logic is presented in this book. The standard form of axiomatic set theory is the zermelofraenkel set theory, together with the axiom of choice. Jan 01, 1974 sets, logic and axiomatic theories book. The benefit of this approach where the general definition of first order logic is developed first is that you apply this to study first order logic in general and other first order theories such that the theory of groups, rings, vector space, random graphs, etc. The set t 2,3,1 is equal to s because they have the same members. Set theory and logic dover books on mathematics kindle edition by stoll, robert r download it once and read it on your kindle device, pc, phones or tablets. It is a quirk of the history of logic that the formalized theories of logic inference that became canonical aimed to regiment singular reference and associated quantifiers and ignored plural reference and plural quantifiers even though we use plural talk in informal maths all the time. The boldest step which cantor had takenin the eyes of his contemporarieswas his use of infinite sets, which he considered as no less natural than using finite sets.
While we usually list the members of a set in a standard order if one is available. Much mathematics can be cleanly and axiomatically developed beginning with axiomatic set theory and then associating axiomatic rules to suitably defined sets and constructive relations. One reason for the increase in importance of mathematical logic was the discovery of paradoxes in naive set theory. Complex issues arise in set theory more than any other area of pure mathematics. The open logic project would like to acknowledge the generous support of thetaylor institute of teaching and learningof the university of calgary, and thealberta open educational re. The standard form of axiomatic set theory is denoted zfc and it consists of the zermelofraenkel zf axioms combined with the axiom of choice c. Axiomatic set theory an overview sciencedirect topics. To generalize this to arbitrary sets aand bwe rst need the concept of an. It is unfortunately true that careless use of set theory can lead to contradictions.
Set theory deals with objects and their collections. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Sets logic and axiomatic theories edition undergraduate. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. The axiomatic system of set theory as we know it today was in. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. In earlier versions of our logic style manual, we tended to omit these. Since this is a set theory text, we reverse the usual procedure by putting in the rst chapter some mathematics that will prove essential to the serious study of set theory. We need it here because the concept of set that the axioms are trying to capture is that of an arbitrary objectinextension and without that. For those that take axiomatic set theory, you will learn about something called bounded comprehension or restricted comprehension. Lucidly and gradually explains sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, boolean algebras, informal axiomatic set theory, several algebraic theories, and 1storder theories.
1653 513 455 673 415 433 1156 1580 1580 418 560 1405 1653 387 1010 1422 1067 1182 963 441 95 409 777 312 677 1130 640 1356 1002 617 346 1189