Ordinals set theory pdf

The real numbers an introduction to set theory and. Cantor as we have seen, the naive use of classes, in particular the connection betweenconceptandextension,ledtocontradiction. Introduction in this chapter, we will look at certain kinds of ordered sets. Set theory basic set theory stanford encyclopedia of. Sets are welldetermined collections that are completely characterized by their elements. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via selfstudy. They are not guaranteed to be comprehensive of the material covered in the course. Thus, two sets are equal if and only if they have exactly the same elements. In set theory, a field of mathematics, the buraliforti paradox demonstrates that constructing the set of all ordinal numbers leads to a contradiction and therefore shows an antinomy in a system that allows its construction. All of these assertions are proved in virtually every book on set theory. Set theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. The axioms of set theory, ordinal and cardinal arithmetic, the axiom of foundation, relativisation, absoluteness, and reflection, ordinal definable sets and inner models of set theory, the constructible universe l cohens method of forcing, independence. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice.

Cantors theory of ordinal and cardinal numbers topics. It is named after cesare buraliforti, who in 1897 published a paper proving a theorem which, unknown to him, contradicted a previously proved result by cantor. We define an nondecreasing nonincreasing sequence of ordinals an ordered set of ordinals where iff. Since is a linear ordering on the ordinals, let be the least ordinal such that but then which is a contradiction. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important. This note is an introduction to the zermelofraenkel set theory with choice zfc. Introduction to modern set theory judith roitman december 6, 2011 1. The axioms of set theory, the natural numbers, the ordinal numbers, relations and orderings, cardinality, there is nothing real about the real numbers, the universe, reflection, elementary submodels and constructibility. The present undergraduate textbook develops the core material on cardinals, ordinals, and the real line. Chapter 2 the zermelo fraenkel axioms of set theory the axiom of extensionality.

This book has been reprinted with the cooperation of. Set theory is indivisible from logic where computer science has its roots. Keywords cantors theorem dedekinds theorem frege zermelofraenkel axiom system order, cardinals, and ordinals set theory, mathematical logic. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. This is enough since you know that a transitive set of ordinals is. A set is called an ordinal i transitive and all 2 are transitive. A book of set theory, first published by dover publications, inc. The class of sets of ordinals which are computable by such machines has strong closure properties and satisfies the set theoretic axiom system so. A countable set is any set that is either finite or the same size as n.

Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Set theory for computer science university of cambridge. Cantorian set theory and its axiomatizations describe a universe of hierarchical sets. Thus, every element of an ordinal is an ordinal and ordinals. Free set theory books download ebooks online textbooks. This is a continuation of my earlier set theory post. If every element of the set a is an element of the set b and every element of the set b is an element of the set a, then a b. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Cantor developed the theory of ordinal numbers which. Pdf theory of sets download full pdf book download. Set theory for beginners consists of a series of basic to intermediate lessons in set theory. Pdf we propose a natural theory so axiomatizing the class of sets of ordinals in a model of zfc set theory. It then delves into contemporary set theory, covering such topics as the borel hierarchy and lebesgue measure.

Basic set theory a set is a many that allows itself to be thought of as a one. Pdf models of set theory with definable ordinals hazem. In fact this definition extends naturally to transfinite ordinals. That is, x is transitive and all of its elements are too. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Suppose x is a set of nonempty sets and is a well ordering on s x. We propose a natural theory so axiomatizing the class of sets of ordinals in a model of zfc set theory. For a more leisurely treatment, see halmos, na ve set theory. In this post, ill describe the next three axioms of zf and construct the ordinal numbers. To understand the philosophical significance of set theory, it will help to have some sense of why set theory arose at all.

A final chapter presents an alternative conception of set theory useful in computer science. Then the set z of elements of x which are ordinals is nonempty, and applying the axiom of foundation to this set we obtain an ordinal. If x is a set of ordinals, the union set s x is also an ordinal. The present book covers each of these areas, giving the reader an understanding of the ideas involved. So, before we get started on discussing set theory at. This is illustrated by the following examples, which are easy to verify from the basic. An acquaintance with various examples and properties of well. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Set theoryordinals wikibooks, open books for an open world. The basic relation in set theory is that of elementhood, or membership.

Pdf models of set theory with definable ordinals ali. Then f y the 2smallest element is a choice function. One of the basic properties of an ordinal is that it is a wellordered set. Set theory with an introduction to real point sets.

Constructibility theory in so corresponds to a natural recursion theory on ordinals. By focusing on the settheoretic aspects of analysis, this text makes the best of two worlds. Jeffrey paris 1973 initiated the study of do models and showed that 1 every consistent extension t of zf has a do. All infinite sets are at least as big as n think of n as the first infinite size. To understand that, it will help to think a little bit about the history and mythology of mathematics. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. Jeffrey paris 1973 initiated the study of do models and showed that. Ordinal addition and multiplication are not commutative.

We can index ordinals with ordinals to generate the notion of a sequence of ordinals. For those of you new to abstract mathematics elementary does not mean simple though much of the material. This chapter investigates ways in which they can be generalized to apply to a very much wider class of ordered sets than the subsets of the basis of this study is the observation that a version of induction can be applied to any ordered set with a property called wellordering. The simple and general principles of induction are powerful tools for proving things about the natural numbers. If a set is ordered in a reasonable way,\ then there is a natural way to define an order topology on. A to indicate that the object a is an element, or a member, of the set a. The spine of the settheoretic universe, and the most essential class of objects in the study of set theory, is the class of ordinals. Set theory some basics and a glimpse of some advanced. An introduction to elementary set theory guram bezhanishvili and eachan landreth 1 introduction in this project we will learn elementary set theory from the original historical sources by two key gures in the development of set theory, georg cantor 18451918 and richard dedekind 18311916.