Ordinal number
Introduction
An ordinal number is a concept in mathematics used to describe the order type of a well-ordered set. Ordinal numbers extend the concept of natural numbers and provide a way to describe the position of an element within a sequence. Unlike cardinal numbers, which describe the size of a set, ordinal numbers focus on the arrangement or order of elements. Ordinal numbers are foundational in set theory and have applications in various branches of mathematics, including topology, analysis, and logic.
Historical Background
The concept of ordinal numbers was first introduced by Georg Cantor, a German mathematician, in the late 19th century. Cantor's work on set theory laid the groundwork for modern mathematical logic and the study of infinite sets. He distinguished between different types of infinities and introduced the notion of ordinal numbers to provide a rigorous framework for understanding the order of elements in infinite sequences.
Formal Definition
In set theory, an ordinal number is defined as a transitive set that is well-ordered by the membership relation. A set \( S \) is transitive if every element of \( S \) is also a subset of \( S \). A set is well-ordered if every non-empty subset has a least element under the given ordering.
Formally, an ordinal number is a set that satisfies the following properties: 1. **Transitivity**: For any elements \( a \) and \( b \) in the set, if \( a \in b \) and \( b \in S \), then \( a \in S \). 2. **Well-Ordering**: Every non-empty subset of the ordinal has a least element.
Types of Ordinal Numbers
Finite Ordinals
Finite ordinals correspond to the natural numbers. They are the simplest form of ordinals and are used to describe the order of elements in a finite sequence. Each finite ordinal is represented by a natural number, and the set of all finite ordinals is denoted by \( \omega \).
Infinite Ordinals
Infinite ordinals extend beyond finite numbers and describe the order type of infinite sequences. The smallest infinite ordinal is \( \omega \), which represents the order type of the natural numbers. Infinite ordinals can be further classified into successor ordinals and limit ordinals.
Successor Ordinals
A successor ordinal is an ordinal that immediately follows another ordinal. If \( \alpha \) is an ordinal, its successor is denoted by \( \alpha + 1 \). Successor ordinals are characterized by having a largest element.
Limit Ordinals
Limit ordinals are ordinals that are not successors of any ordinal. They do not have a largest element and are limits of all smaller ordinals. The ordinal \( \omega \) is an example of a limit ordinal, as it is the limit of all finite ordinals.
Arithmetic of Ordinals
Ordinal arithmetic is a set of operations defined on ordinal numbers. The primary operations are addition, multiplication, and exponentiation. These operations differ from their counterparts in cardinal arithmetic due to the order-sensitive nature of ordinals.
Ordinal Addition
Ordinal addition is defined recursively. For ordinals \( \alpha \) and \( \beta \), the addition \( \alpha + \beta \) is the order type of the disjoint union of two well-ordered sets of order types \( \alpha \) and \( \beta \), with all elements of the first set preceding those of the second.
Ordinal Multiplication
Ordinal multiplication is also defined recursively. For ordinals \( \alpha \) and \( \beta \), the multiplication \( \alpha \cdot \beta \) is the order type of the Cartesian product of two well-ordered sets of order types \( \alpha \) and \( \beta \), ordered lexicographically.
Ordinal Exponentiation
Ordinal exponentiation is defined using transfinite recursion. For ordinals \( \alpha \) and \( \beta \), the exponentiation \( \alpha^\beta \) is the order type of the set of functions from \( \beta \) to \( \alpha \) with finite support, ordered lexicographically.
Ordinal Notations
Ordinal numbers can be represented using various notations. The most common notation is the Cantor Normal Form, which expresses ordinals as sums of decreasing powers of \( \omega \) with natural number coefficients. This notation provides a unique representation for each ordinal and is useful for performing ordinal arithmetic.
Applications of Ordinals
Ordinal numbers have numerous applications in mathematics and related fields. They are used in the study of well-ordered sets, transfinite induction, and recursion. Ordinals also play a crucial role in the theory of cardinal numbers, where they are used to define cardinality and compare the sizes of infinite sets.
In topology, ordinals are used to define ordinal spaces, which are topological spaces with order topology. These spaces are important in the study of convergence and compactness.
In logic, ordinals are used in the study of ordinal definable sets and in the analysis of formal systems. They are also used in proof theory to measure the strength of mathematical theories.