In these notes, we will discuss set theory using the language of zfc. In set theory, with the study of the ordinal numbers, ones intuition for the real numbers ought to be an advantage, even as, with some new hypotheses, one goes on to create a new world. Jul 02, 20 set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions number, order, and function, taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis. Thus nis a set with the property that each of its elements is simultaneously an element of n and a subset of n. 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. Ordinals and cardinals live happily within standard set theory.
On the other hand, a set is said to be denumerable or denumerably infinite if it is equipollent to the set. Ordinal numbers and predicative set theory godelas predicative model of classical set theory relative t o classical ordinals and more recent works by many logicians on hyperarithmetic sets produce the impression that the question of making the philosophical notion of predicativity more exact has become more nearly manageable. 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. In this formalism one may write down the above definition of r, and see that it. Note that his an ordinal, and we often denote this particular ordinal by 0. By naming the each mathematical concept constructed by the set of zfc set theory as, for example, the numbers 0, 1, 2, or the function f, etc. Definition of exponentiation of ordinals definition of functions from one wellordered set to another. Moreover, every previous natural number is a subset of n.
Difference between cardinal numbers and ordinal numbers. 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. The concept of the set of all ordinal numbers is contradictory. In such a course of study one first covers the basics, including cardinal and ordinal numbers and the methods of proof and definition by induction. These new numbers also have an arithmetic that extends that 5. Chapter 2 the zermelo fraenkel axioms of set theory the axiom of extensionality. In set theory, an ordinal number or ordinal is an equivalence class of wellordered sets under the relation of order isomorphism. Browse other questions tagged set theory ordinal numbers metamathematics or ask your own question. The set x is said to be an ordinal if x is transitive and the. Zermelos axiomatization of set theory stanford encyclopedia. We give certain definitions and results from the theory of ordinals. Introduction there are numerous mathematical generalizations of the natural numbers. The order types ordinal numbers themselves are wellordered in a natural way, and this wellordering must have an order type \omega.
You need to distinguish between the ordinal numbers and a particular representation in set theory, which is akin to the distinction between a number and a numeral in arithmetic. On wellorderable sets, this cardinality function satisfies, but one needs the axiom of choice precisely, the wellordering theorem to prove that every set is wellorderable. Then f y the 2smallest element is a choice function. Since nobody likes to work in a theory where one has to carry around not only the. We define number of elements in terms of sameness of. Informally, an ordinal number is the ordertype of a wellordered set.
The person then simply counts those objects, and gives the ordinal numbers to them. The definition of implies in zf, zermelofraenkel set theory without the axiom of choice that no cardinal number is between and. Cardinals measure the size of a set in correlation sense. The original definition of ordinal numbers, found for example in the principia mathematica, defines the order type of a wellordering as the set of all wellorderings similar orderisomorphic to that wellordering. An ordinal number is commonly used in english when describing the relationship of natural numbers.
To define the real numbers as dedekind cuts of the rational num bers, we take, by the power set axiom pow a set y such that. Specifically, ordinal numbers generalise the concept of the next number after or the index of the next item after. The situation for set theory is somewhat different. We shall now give a formal definition of ordinal numbers.
Indeed, the concept of ordinal numbers emerged from georg cantors work on set theory together with the concept of cardinal numbers. So y is the set of all ordinals, y is a transitive set of ordinals, so it is an ordinal itself. This needs to be done in a formalized setting, based on a system of axioms, if one. Ordinal number simple english wikipedia, the free encyclopedia. Predicative definition and existence of ordinal numbers. A binary relation on a set \a\ is a set of ordered pairs of elements of \a\, that is, a. Indeed, if it were a set, it would itself be an ordinal number, so it would have to contain itself as an element. It is handy as it can represent ordinals as actual objects within a set theory. Euclid and archimedes at the beginning of heibergs edition of the elements, some hypotheses are written out in the form of postulates. And, like cardinals, ordinals are also natural numbers.
In set theory, an ordinal number, or just ordinal, is the order type of. Set symbols of set theory and probability with name and definition. A limit number is an ordinal which is a limit element of the ordinals. These notes for a graduate course in set theory are on their way to be coming a book. For your point 1, the ordinals of set theory are intended to capture all we mean by first, second, third, and then extend the concept in a welldefined way into the infinite. In set theory, ordinals have meaning only in relation to sets that are wellordered.
The concept of ordinal numbers and the concept of natural numbers are constructed by such a way. Here is the definition used in zermelo fraenkel set theory zf. The collection of ordinal numbers is too big to be a set. It is named after cesare buraliforti, who in 1897 published a paper proving a theorem which, unknown to him, contradicted a previously proved result. It is easily shown in naive set theory and remains true in zfc but not in new foundations that the order type of all ordinal numbers less than a fixed \alpha is \alpha itself. Accordingly, the set of all ordinal numbers does not exist. Specifically, a set is said to be an ordinal if and only if is wellordered by the membership relation and is transitive, that is, every element in is a subset of. Ordinals are an extension of the natural numbers different from. Thus the ordinal number wc is a measure of the intrinsic logical strength of the hilbert basis theorem. Suppose x is a set of nonempty sets and is a well ordering on s. The equivalence is provable in the weak base theory rcao. 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 any finite collection of objects can be put in order just by the process of counting.
An introduction to set theory university of toronto. The number zero is the only natural number that cannot be an ordinal number. On the one hand, there are the integers, the rational numbers, the real numbers, and the complex numbers. Lesson 63 worksheet 3 number sense name ordinal numbers 1st 2 label the cars from 1st to 5th. There are a number of other operations that go along with the set theory as well. Successor ordinals are named as such because they are the immediate successor of some other ordinal. Buraliforti paradox project gutenberg selfpublishing.
Suppose x is a set of nonempty sets and is a well ordering on s x. 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. Addition of ordinals definition of concatenation of wellordered sets. A set is called an ordinal i transitive and all 2 are transitive. In base 2 we use the numeral 110012 to represent this number. In set theory, an ordinal number, or just ordinal, is the order type of a wellordered set.
Feb 20, 2020 the ordinal numbers or just ordinals constitute a generalisation of a natural numbers to numbers of possibly infinite magnitudes. A set is finite iff it is equipollent to some element of otherwise, it is infinite. Ordered position, ordinal position, and ordinality are terms that refer to the place. Set theory basic set theory stanford encyclopedia of. The set theory also makes it possible to include the number zero as a natural number. How are koepkes ordinal computability and erecursion related. In contemporary set theory one usually defines the ordinals so that, for each ordinal, pred. Cantor developed the theory of ordinal numbers which 80 extends the standard natural numbers n. Ordinal numbers and predicative set theory ordinal numbers and predicative set theory wang, hao 19590101 00. It is hoped that the reader will eventually perceive the ordinal. Contents 1 introduction 2 2 the zermelo fraenkel axioms of set theory 5 3 ordinals 14. By his definition the ordinal number one is the class of all wellordered singleton sets, the ordinal number two is the class of all wellordered doubleton sets, etc. Prove that a set is an ordinal number iff it is a transitive set of transitive sets. If nis a natural number, then all previous natural numbers are elements of n.
Complex issues arise in set theory more than any other area of pure mathematics. Recall from last lecture that a set xis an ordinal just in case it is transitive and wellordered by pmembership. Jun 03, 1997 a cardinal number is one way to measure the size of a set. A set is an ordinal number or an ordinal if it is transitive and wellordered by for ordinals. We define the set e of notations for ordinals less than es, and the ordering definition is given by the following inductive. Ordinal numbers and the hilbert basis theorem authors. They are usually identified with hereditarily transitive sets.
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. Following the literature from the origin of set theory in the late 19th century to more current times, an arithmetic of finite and transfinite ordinal numbers is outlined. Ordinal definable sets were introduced by godel 1965. We define the ordinal number of u to be the image ordu df vu. Ordinal numbers and predicative set theory, mathematical. On the other hand, there are the cardinal numbers and the ordinal numbers. One approach to representing ordinal numbers is to define an ordinal number to be an equivalence class of well ordered sets under the equivalence relation nduced by order isomorphism. In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal.
Cardinal and ordinal numbers math 6300 klaus kaiser april 9, 2007. Formalizations of the matchstick diagram representation of ordinals. The cardinal or cardinality of x, denoted cardx, is. An ordinal number or ordinal is a number that indicates position or order of objects, such as first, second, third, and so on figure 1. This is itself an ordinal number larger than all countable ones, so it is an uncountable set.
In set theory, ordinals are also ordinal numbers people use to order infinite sets. I was thinking about ordinal numbers recently, after i have read the wiki article about impredicativity. The concept of a set is outlined and directed to the understanding that an ordinal, a special kind of number, is a particular kind of well ordered set. Overall, the ordinal numbers are very complicated, but they can be quite helpful when it comes to trans nite recursion and induction. Well use natural number to mean ordinal which can be constructed by. In set theory, an ordinal number, or ordinal, is one. The natural number system can be extended to the system of ordinal num bers. 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.
Two sets a and b are said to have the same cardinality, and. This works in some settings, but in zfc set theory it has the problem that these equivalence classes will not be sets, except for the ordinal 0 only the empty. Apr 02, 2020 a cardinal number is then any cardinality, i. We assume a standard axiomatization of set theory, e. Direct axiomatization of ordinal and cardinal numbers. Intuitively speaking, the ordinals form a number system that can be viewed as an extension of the natural numbers into infinite values. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. A set t is transitive if every element of t is a subset of t. Definition of ordinals in set theory in layman terms. There are several approaches to define ordinal numbers, some of which are considered below.
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