The finite ordinal numbers are those obtained by starting with \\varnothing\ and repeatedly taking the successor. In set theory the natural numbers are defined as the finite ordinals. Set theory basic set theory stanford encyclopedia of. Short notes on axioms of set theory, well orderings and ordinal numbers august 29, 20. This theory will prove to be easy to manage and fairly powerful in the following sense. Halmos was quite painful to work through, because there was little mathematical notation.
Suppose x is a set of nonempty sets and is a well ordering on s x. Clearly one can obtain a hierarchy by repeatedly forming the powerset of a powerset as follows. 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. It is hoped that the reader will eventually perceive the ordinal. We list the axioms of set theory, for the moment without the the axiom of regularity.
Settheoretic definition of natural numbers wikipedia. The content of the rst equivalence in 1 is then that every ordinal is the set of its predecessors, just as for natural numbers. Then f y the 2smallest element is a choice function. Introduction there are numerous mathematical generalizations of the natural numbers. Pdf theory of sets download full pdf book download. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. I later discovered endertons elements of set theory and i rec.
It then delves into contemporary set theory, covering such topics as the borel hierarchy and lebesgue measure. In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a possibly infinite collection of objects in order, one after another. The ordinal 0 is defined to be the empty set the ordinal 1 is defined to be the set, which is of course equal to. See the contents of the notes given below to see the topics covered by. This paper will present a brief settheoretic construction of the natural numbers before discussing in detail the ordinal and cardinal numbers. Abbie rossman ordinal numbers lesson plan 1 lesson plan introduction lesson topic ordinal numbers 1st through 10th length of lesson 90. This note is an introduction to the zermelofraenkel set theory with choice zfc. Cantors theory of ordinal and cardinal numbers topics.
A book of set theory, first published by dover publications, inc. A countable set is any set that is either finite or the same size as n. 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. A taste of set theory for philosophers helsingin yliopisto. In mathematical set theory, a set s is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a firstorder formula. A book of set theory dover books on mathematics paperback july 16, 2014.
The present book covers each of these areas, giving the reader an understanding of the ideas involved. The proofs of theorems files were prepared in beamer and they contain proofs of results which are particularly lengthy shorter proofs are. In contrast to cardinal numbers, which serve to designate the size of a set, ordinal numbers are used to determine positions within a prescribed wellordered sequence. Introduction to set theory notes introduction to set theory, second edition revised and expanded, by karel hrbacuk and thomas jech, dekker 1984 copies of the classnotes are on the internet in pdf format as given below. Pdf we propose a natural theory so axiomatizing the class of sets of ordinals in a model of zfc set theory. Remember that in set theory, we define natural numbers as follows. Since supais again an ordinal, it is just the set of all ordinals smaller than it.
A set is pure if all of its members are sets, all members of its members are sets, and so on. I worked my way through halmos naive set theory, and did about of robert vaughts book. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. The set n of natural numbers is defined in this system as the smallest set containing 0 and closed under the successor function s defined by sn n. The structure n, 0, s is a model of the peano axioms. Set theory the third millennium edition, revised and. Anwar khan for providing these notes and appreciates his effort to publish these notes on it covers the complete syllabus of advanced analysis paper of msc mathematics. Weaklyinaccessible ordinal numbers allow a classification similar to the classification of inaccessible cardinals cf. Contents the natural numbers the university of chicago. He defined a first infinite ordinal number which comes after all the natural numbers n and which he called after.
All infinite sets are at least as big as n think of n as the first infinite size. Ordinal numbers set theory, naively or axiomatically, is a natural, strong and convenient theory which can be used as a foundation for all of mathematics. Set theoryordinals wikibooks, open books for an open world. A final chapter presents an alternative conception of set theory useful in computer science.
Chapter 2 the zermelo fraenkel axioms of set theory the axiom of extensionality. Set theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. Schema for transfinite induction and ordinal arithmetic. The sum and the product of two ordinal numbers is an ordinal number. The axiom of pair, the axiom of union, and the axiom of. Cardinality of sets 16 acknowledgments 17 references 18 1. In axiomatic set theory we assume that there is a mathematical structure u which we call the universe and whose elements are called sets. On the one hand, there are the integers, the rational numbers, the real numbers, and the complex numbers. The continuum hypothesis its natural to ask if there is a set thats larger than the set of natural numbers. So y is the set of all ordinals, y is a transitive set of ordinals, so it is an ordinal itself. Any finite collection of objects can be put in order just by the process of counting.
Set symbols of set theory and probability with name and definition. If a is a sentence of the theory of ordinal numbers, then a is a theorem of our system if and only if the natural translation of a in set theory is a theorem of zermelofraenkel set theory. But this definition has a serious defect from the point of view of zf set theory because the class of all singleton sets is a proper class, as is the class of all doubleton sets, etc. Cardinality is studied for its own sake as part of set theory. Constructibility theory in so corresponds to a natural recursion theory on ordinals. Thus, the successor of \\alpha\ is just the set \\alpha\ together with one more element, namely, \\alpha\ itself. Characterizations of ordinal numbers in set theory. Short notes on axioms of set theory, well orderings and.
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