Ordinals set theory pdf

The class of sets of ordinals which are computable by such machines has strong closure properties and satisfies the set theoretic axiom system so. That is, x is transitive and all of its elements are too. Cantor as we have seen, the naive use of classes, in particular the connection betweenconceptandextension,ledtocontradiction. To understand the philosophical significance of set theory, it will help to have some sense of why set theory arose at all. Set theory basic set theory stanford encyclopedia of. To understand that, it will help to think a little bit about the history and mythology of mathematics. Then f y the 2smallest element is a choice function.

Cantors theory of ordinal and cardinal numbers topics. Thus, every element of an ordinal is an ordinal and ordinals. The present book covers each of these areas, giving the reader an understanding of the ideas involved. Pdf theory of sets download full pdf book download. 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. So, before we get started on discussing set theory at. The spine of the settheoretic universe, and the most essential class of objects in the study of set theory, is the class of ordinals. Jeffrey paris 1973 initiated the study of do models and showed that. Basic set theory a set is a many that allows itself to be thought of as a one. 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. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. A set is called an ordinal i transitive and all 2 are transitive. This note is an introduction to the zermelofraenkel set theory with choice zfc. Pdf we propose a natural theory so axiomatizing the class of sets of ordinals in a model of zfc set theory.

This book has been reprinted with the cooperation of. An acquaintance with various examples and properties of well. 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. Chapter 2 the zermelo fraenkel axioms of set theory the axiom of extensionality. They are not guaranteed to be comprehensive of the material covered in the course. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. By focusing on the settheoretic aspects of analysis, this text makes the best of two worlds. Pdf models of set theory with definable ordinals hazem. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student.

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. All of these assertions are proved in virtually every book on set theory. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Since is a linear ordering on the ordinals, let be the least ordinal such that but then which is a contradiction. 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. Free set theory books download ebooks online textbooks. For those of you new to abstract mathematics elementary does not mean simple though much of the material. 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. A final chapter presents an alternative conception of set theory useful in computer science. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important. For a more leisurely treatment, see halmos, na ve set theory. Suppose x is a set of nonempty sets and is a well ordering on s x. 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.

Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. The basic relation in set theory is that of elementhood, or membership. Set theory is indivisible from logic where computer science has its roots. We propose a natural theory so axiomatizing the class of sets of ordinals in a model of zfc set theory. One of the basic properties of an ordinal is that it is a wellordered set.

This is a continuation of my earlier set theory post. Set theory for computer science university of cambridge. Set theory with an introduction to real point sets. Introduction to modern set theory judith roitman december 6, 2011 1. If x is a set of ordinals, the union set s x is also an ordinal. If a set is ordered in a reasonable way,\ then there is a natural way to define an order topology on. 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. Set theory for beginners consists of a series of basic to intermediate lessons in set theory. Ordinal addition and multiplication are not commutative. Sets are welldetermined collections that are completely characterized by their elements. Constructibility theory in so corresponds to a natural recursion theory on ordinals.

The real numbers an introduction to set theory and. A book of set theory, first published by dover publications, inc. All infinite sets are at least as big as n think of n as the first infinite size. See, for example, jech, set theory, for a clear, concise presentation in just a few pages. Keywords cantors theorem dedekinds theorem frege zermelofraenkel axiom system order, cardinals, and ordinals set theory, mathematical logic.

Thus, two sets are equal if and only if they have exactly the same elements. The simple and general principles of induction are powerful tools for proving things about the natural numbers. Set theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. In fact this definition extends naturally to transfinite ordinals. We define an nondecreasing nonincreasing sequence of ordinals an ordered set of ordinals where iff.

Pdf models of set theory with definable ordinals ali. The main notions of set theory cardinals, ordinals, transfinite induction are fundamental to all mathematicians, not only to those who specialize in mathematical logic or settheoretic topology. Set theoryordinals wikibooks, open books for an open world. Cantorian set theory and its axiomatizations describe a universe of hierarchical sets. Jeffrey paris 1973 initiated the study of do models and showed that 1 every consistent extension t of zf has a do. This is enough since you know that a transitive set of ordinals is. 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. A countable set is any set that is either finite or the same size as n. We can index ordinals with ordinals to generate the notion of a sequence of ordinals. A do model here also referred to a paris model is a model of set theory all of whose ordinals are first order definable in. This is illustrated by the following examples, which are easy to verify from the basic. 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. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice.