Ordinal number

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Ordinal numbers are a fundamental concept in mathematics, particularly in the fields of set theory and order theory. They are used to describe the position of an element within a well-ordered set, such as first, second, third, and so on. Unlike cardinal numbers, which quantify the number of elements in a set, ordinal numbers focus on the order of elements.

Definition

In formal terms, an ordinal number is defined based on the concept of transfinite numbers, introduced by Georg Cantor. An ordinal number represents the order type of a well-ordered set. This means it characterizes the abstract relationships of being "before" or "after" within a set, without reference to the specific nature of the elements of the set.

Notation

Ordinal numbers are often denoted using lowercase Greek letters, such as α (alpha), β (beta), and γ (gamma). The first ordinal number is 0, which represents the empty set. The next ordinal is 1, representing a set with one element, and so on for the finite ordinals. After all finite ordinals, comes the first infinite ordinal, denoted ω (omega), which represents the order type of the set of all natural numbers.

Properties

Ordinal numbers have several important properties:

• Transitivity: If α is an ordinal and β is a member of α, then β is also an ordinal and every element of β is an element of α.
• Well-ordered: Every non-empty set of ordinals has a least element under the usual "<" (less than) relation.
• Successor and Limit Ordinals: Every ordinal has a successor, which is the smallest ordinal greater than it. Ordinals that are not successors are called limit ordinals, with ω being the first limit ordinal.

Applications

Ordinal numbers have wide applications in various branches of mathematics:

• In set theory, they are used to describe the sizes of well-ordered sets.
• In model theory, ordinals are used in the construction of models.
• In proof theory and theoretical computer science, ordinals are used to measure the complexity of proofs and algorithms.

See Also

• Cardinal number
• Transfinite number
• Aleph number
• Well-ordered set

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