# Ordinal number

*transfinite number, ordinal*

The order type of a well-ordered set. This notion was introduced by G. Cantor in 1883 (see [2]). For instance, the ordinal number of the set $ \mathbf{N} $ of all positive integers, ordered by the relation $ \leq $, is $ \omega $. The ordinal number of the set consisting of $ 1 $ and numbers of the form $ 1 - \dfrac{1}{n} $ where $ n \in \mathbf{N} $, ordered by the relation $ \leq $, is $ \omega + 1 $. One says that an ordinal number $ \alpha $ is **equal to** (**less than**) an ordinal number $ \beta $, written $ \alpha = \beta $ ($ \alpha < \beta $), if and only if a set of type $ \alpha $ is similar to (a proper segment of) a set of type $ \beta $. For arbitrary ordinal numbers $ \alpha $ and $ \beta $, one and only one of the following possibilities holds:

- $ \alpha < \beta $.
- $ \alpha = \beta $.
- $ \alpha > \beta $.

The set $ \{ \beta \mid \beta < \alpha \} $ of all ordinal numbers less than $ \alpha $ is well-ordered with type $ \alpha $ by the relation $ \leq $. Moreover, any set of ordinal numbers is well-ordered by the relation $ \leq $, i.e., any non-empty set of ordinal numbers contains a least ordinal number. For any set $ Z $ of ordinal numbers, there exists an ordinal number greater than any ordinal number from $ Z $. Accordingly, the set of all ordinal numbers does not exist. The smallest of the ordinal numbers following an ordinal number $ \alpha $ is called the **successor** of $ \alpha $ and is denoted by $ \alpha + 1 $. The ordinal number $ \alpha $ is called the **predecessor** of the ordinal number $ \alpha + 1 $. An ordinal number is called a **limit** ordinal number if and only if it does not have a predecessor. Thus, $ 0 $ is a limit ordinal number. Any ordinal number can be represented in the form $ \alpha = \lambda + n $, where $ \lambda $ is a limit ordinal number and $ n $ is an integer, the sum being understood in the sense of addition of order types.

A **transfinite sequence** of type $ \alpha $, or an **$ \alpha $-sequence**, is a function $ \phi $ defined on $ \{ \beta \mid \beta < \alpha \} $. If the values of this sequence are ordinal numbers, and if $ \gamma < \beta < \alpha $ implies that $ \phi(\gamma) < \phi(\beta) $, then it is called an **ascending sequence**. Let $ \phi $ denote a $ \lambda $-sequence, where $ \lambda $ is a limit ordinal number. The least of the ordinal numbers greater than any $ \phi(\gamma) $, where $ \gamma < \lambda $, is called the **limit** of the sequence $ (\phi(\gamma))_{\gamma < \lambda} $ and is denoted by $ \displaystyle \lim_{\gamma < \lambda} \phi(\lambda) $. For instance, $ \displaystyle \omega = \lim_{n < \omega} n = \lim_{n < \omega} n^{2} $. An ordinal number $ \lambda $ is **cofinal** to a limit ordinal number $ \alpha $ if and only if $ \lambda $ is the limit of an ascending $ \alpha $-sequence: $ \displaystyle \lambda = \lim_{\xi < \alpha} \phi(\xi) $. The ordinal number $ \mathsf{cf}(\lambda) $ is the least ordinal number to which $ \lambda $ is cofinal.

An ordinal number is called **regular** if and only if it is not cofinal to any smaller ordinal number, otherwise it is called **singular**. An infinite ordinal number is called an **initial** ordinal number of cardinality $ \tau $ if and only if it is the least among the ordinal numbers of cardinality $ \tau $ (i.e., among the order types of well-ordered sets of cardinality $ \tau $). Hence, $ \omega $ is the least initial ordinal number. The initial ordinal number of power $ \tau $ is denoted by $ \omega(\tau) $. The set $ \{ \omega(\delta) \mid \aleph_{0} \leq \delta < \tau \} $ of all initial ordinal numbers of infinite cardinality less than $ \tau $ is well-ordered. If the ordinal number $ \alpha $ is its order type, then one puts $ \omega(\tau) = \omega_{\alpha} $. Therefore, every initial ordinal number is provided with an index equal to the order type of the set of all initial ordinal numbers less than it. In particular, $ \omega_{0} = \omega $. Different indices correspond to different initial ordinal numbers. Each ordinal number $ \alpha $ is the index of some initial ordinal number. If $ \lambda $ is a limit ordinal number, then $ \mathsf{cf}(\lambda) $ is a regular initial ordinal number.

An initial ordinal number $ \omega_{\alpha} $ is called **weakly inaccessible** if and only if it is regular and its index $ \alpha $ is a limit ordinal number. For instance, $ \omega = \omega_{0} $ is weakly inaccessible, but $ \omega_{\omega} $ is singular and is thus not weakly inaccessible. If $ \alpha > 0 $, then $ \omega_{\alpha} $ is weakly inaccessible if and only if $ \alpha = \omega_{\alpha} = \mathsf{cf}(\alpha) $.

Weakly-inaccessible ordinal numbers allow a classification similar to the classification of inaccessible cardinal numbers. The sum and the product of two ordinal numbers is an ordinal number. If the set of indices is well-ordered, then the well-ordered sum of ordinal numbers is an ordinal number. One can also introduce the operation of raising to a power, by transfinite induction:

- $ \gamma^{0} \stackrel{\text{df}}{=} 1 $.
- $ \gamma^{\xi + 1} \stackrel{\text{df}}{=} \gamma^{\xi} \cdot \gamma $.
- $ \displaystyle \gamma^{\lambda} \stackrel{\text{df}}{=} \lim_{\xi < \lambda} \gamma^{\xi} $, where $ \lambda $ is a limit ordinal number.

The number $ \gamma^{\alpha} $ is called a **power** of a number $ \gamma $, where $ \gamma $ is called the **base** of the power and $ \alpha $ the **exponent** of the power. For example, if $ \gamma = \omega $ and $ \alpha_{0} = 1 $, then one obtains
$$
\alpha_{1} = \gamma^{\alpha_{0}}, \quad \alpha_{2} = \omega^{\omega}, \quad \alpha_{3} = \omega^{\omega^{\omega}}, \quad \ldots.
$$
The limit of this sequence, $ \displaystyle \epsilon \stackrel{\text{df}}{=} \lim_{n < \omega} \alpha_{n} $, is the least critical number of the function $ \xi \mapsto \omega^{\xi} $, i.e., the least ordinal number $ \alpha $ among those for which $ \omega^{\alpha} = \alpha $. Numbers $ \alpha $ for which this equality holds are called **epsilon-ordinals**.

Raising to a power can be used to represent ordinal numbers in a form resembling the decimal representation of positive integers. If $ \gamma > 1 $ and $ 1 \leq \alpha < \gamma^{\eta} $, then there exists a positive integer $ n $ and sequences $ \beta_{1},\ldots,\beta_{n} $ and $ \eta_{1},\ldots,\eta_{n} $ such that
\begin{gather}
\alpha = \gamma^{\eta_{1}} \cdot \beta_{1} + \cdots + \gamma^{\eta_{n}} \cdot \beta_{n}, \qquad (1) \\
\eta > \eta_{1} > \ldots > \eta_{n}, \qquad 0 \leq \beta_{i} < \gamma, \qquad (2)
\end{gather}
for $ i \in \{ 1,\ldots,n \} $. Formula (1) for the numbers $ \beta_{j} $ and $ \eta_{j} $ satisfying the conditions in (2) is called the **representation** of the ordinal number $ \alpha $ in the base $ \gamma $. The numbers $ \beta_{i} $ are called the **digits**, and the numbers $ \eta_{i} $ the **exponents** of this representation. The representation of an ordinal number in a given base is unique. The representation of ordinal numbers in the base $ \omega $ is used to define the natural sum and the natural product of ordinal numbers.

#### References

[1] | P.S. Aleksandrov, “Einführung in die Mengenlehre und die Theorie der reellen Funktionen”, Deutsch. Verlag Wissenschaft. (1956). (Translated from Russian) |

[2] | G. Cantor, “Contributions to the founding of the theory of transfinite numbers”, Dover, reprint (1952). (Translated from German) |

[3] | F. Hausdorff, “Grundzüge der Mengenlehre”, Leipzig (1914). (Reprinted (incomplete) English translation: Set theory, Chelsea (1978)). |

[4] | K. Kuratowski, A. Mostowski, “Set theory”, North-Holland (1968). |

[5] | W. Sierpiński, “Cardinal and ordinal numbers”, PWN (1958). |

#### Comments

The ordinal $ \mathsf{cf}(\lambda) $, the least ordinal number to which $ \lambda $ is cofinal, is called the **cofinality** of $ \lambda $.

The ordinal number $ \omega $ and (by the axiom of choice) each initial ordinal number with a successor-index are regular. Initial ordinal numbers with a limit-index are singular in general. More precisely, if the axioms of $ \mathsf{ZF} $ are consistent, they remain so after the addition of the axiom that states that all initial ordinal numbers with limit-index $ > 0 $ are singular. Therefore, the axioms of $ \mathsf{ZF} $, if consistent, cannot prove that there are any weakly-inaccessible ordinal numbers other than $ \omega $.

For countable ordinal numbers, see also Descriptive set theory.

#### References

[a1] | K. Kuratowski, “Introduction to set theory and topology”, Pergamon (1972). (Translated from Polish) |

[a2] | T.J. Jech, “Set theory”, Acad. Press (1978). (Translated from German) |

[a3] |
J. Barwise (ed.), Handbook of mathematical logic, North-Holland (1977). (Especially the article of D.A. Martin on Descriptive set theory). |

[a4] | A. Levy, “Basic set theory”, Springer (1979). |

**How to Cite This Entry:**

Ordinal number.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Ordinal_number&oldid=40148