Ordinal Number

In informal usage, an ordinal number is an adjective which describes the numerical position of an object, e.g., first, second, third, etc.

In technical mathematics, an ordinal number is one of the numbers in Georg Cantor's extension of the Whole Numbers. The ordinal numbers are 0, 1, 2, ..., $\omega$, $\omega+1$, $\omega+2$, ..., $\omega+\omega$, $\omega+\omega+1$, .... Cantor's ``smallest'' Transfinite Number $\omega$ is defined to be the earliest number greater than all Whole Numbers, and is denoted by Conway and Guy (1996) as $\omega=\{0, 1, \dots\vert\}$. The notation of ordinal numbers can be a bit counterintuitive, e.g., even though $1+\omega=\omega$, $\omega+1>\omega$.

Ordinal numbers have some other rather peculiar properties. The sum of two ordinal numbers can take on two different values, the sum of three can take on five values. The first few terms of this sequence are 2, 5, 13, 33, 81, 193, 449, $33^2$, $33\cdot 81$, $81^2$, $81\cdot 193$, $192^2$, ... (Conway and Guy 1996, Sloane's A005348). The sum of $n$ ordinals has either $193^a81^b$ or $33\cdot 81^a$ possible answers for $n\geq 15$ (Conway and Guy 1996).

$r\times\omega$ is the same as $\omega$, but $\omega\times r$ is equal to . $\omega^2$ is larger than any number of the form $\omega\times r$, $\omega^3$ is larger than $\omega^2$, and so on.

There exist ordinal numbers which cannot be constructed from smaller ones by finite additions, multiplications, and exponentiations. These ordinals obey Cantor's Equation. The first such ordinal is

$$\epsilon_0=\underbrace{{\omega}^{{\omega}^{\cdot^{\cdot^{\cd... ...{\omega}=1+\omega+\omega^\omega+\omega^{\omega^\omega}+\ldots.$$

The next is

$$\epsilon_1=(\epsilon_0+1)+\omega^{\epsilon_0+1}+{\omega^\omega}^{\epsilon_0+1}+\ldots,$$

then follow $\epsilon_2$, $\epsilon_3$, ..., $\epsilon_\omega$, $\epsilon_{\omega+1}$, ..., $\epsilon_{\omega\times 2}$, ..., $\epsilon_{\omega^2}$, $\epsilon_{\omega^\omega}$, ..., $\epsilon_{\epsilon_0}$, $\epsilon_{\epsilon_0+1}$, ..., $\epsilon_{\epsilon_0+\omega}$, ..., $\epsilon_{\epsilon_0+\omega^\omega}$, ..., $\epsilon_{\epsilon_0\times 2}$, ..., $\epsilon_{\epsilon_1}$, ..., $\epsilon_{\epsilon_2}$, ..., $\epsilon_{\epsilon_\omega}$, ..., $\epsilon_{\epsilon_{\epsilon_0}}$, ..., $\epsilon_{\epsilon_{\epsilon_1}}$, ..., $\epsilon_{\epsilon_{\epsilon_\omega}}$, ..., , ... (Conway and Guy 1996).

See also Axiom of Choice, Cantor's Equation, Cardinal Number, Order Statistic, Power Set, Surreal Number

References

Cantor, G. Über unendliche, lineare Punktmannigfältigkeiten, Arbeiten zur Mengenlehre aus dem Jahren 1872-1884. Leipzig, Germany: Teubner-Archiv zur Mathematik, 1884.

Conway, J. H. and Guy, R. K. ``Cantor's Ordinal Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 266-267 and 274, 1996.

Sloane, N. J. A. Sequence A005348/M1435 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.