View source for Ordinal number

''transfinite number, ordinal''

The [[Order type|order type]] of a [[Well-ordered set|well-ordered set]]. This notion was introduced by G. Cantor in 1883 (see [[#References|[2]]]). For instance, the ordinal number of the set of all positive integers ordered by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o0701801.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o0701802.png" />. The ordinal number of the set consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o0701803.png" /> and of the numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o0701804.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o0701805.png" /> ordered by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o0701806.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o0701807.png" />. One says that an ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o0701808.png" /> is equal to (less than) an ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o0701809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018011.png" />) if a set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018012.png" /> is similar to (a proper segment of) a set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018013.png" />. For arbitrary ordinal numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018015.png" /> one and only one of the following possibilities holds: either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018016.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018017.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018018.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018019.png" /> of all ordinal numbers less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018020.png" /> is well-ordered with type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018021.png" /> by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018022.png" />. Moreover, any set of ordinal numbers is well-ordered by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018023.png" />, i.e. any non-empty set of ordinal numbers contains a least ordinal number. For any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018024.png" /> of ordinal numbers there exists an ordinal number greater than any ordinal number from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018025.png" />. Accordingly, the set of all ordinal numbers does not exist. The smallest of the ordinal numbers following an ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018026.png" /> is called the successor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018027.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018028.png" />. The ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018029.png" /> is called the predecessor of the ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018030.png" />. An ordinal number is called a limit (ordinal) number if it does not have a predecessor. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018031.png" /> is a limit number. Any ordinal number can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018033.png" /> is a limit number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018034.png" /> is an integer, the sum is understood in the sense of addition of order types (cf. [[Order type|Order type]]).

A transfinite sequence of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018035.png" />, or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018037.png" />-sequence, is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018038.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018039.png" />. If the values of this sequence are ordinal numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018040.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018041.png" />, then it is called an ascending sequence. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018042.png" /> denote a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018043.png" />-sequence, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018044.png" /> is a limit number. The least of the ordinal numbers greater than any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018046.png" />, is called the limit of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018047.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018048.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018049.png" />. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018050.png" />. An ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018051.png" /> is cofinal to a limit number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018052.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018053.png" /> is the limit of an ascending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018054.png" />-sequence: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018055.png" />. The ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018056.png" /> is the least ordinal number to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018057.png" /> is cofinal.

An ordinal number is called regular 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018058.png" /> if it is the least among the ordinal numbers of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018059.png" /> (i.e. among the order types of well-ordered sets of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018060.png" />). Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018061.png" /> is the least initial number. The initial ordinal number of power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018062.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018063.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018064.png" /> of all initial ordinal numbers of infinite cardinality less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018065.png" /> is well-ordered. If the ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018066.png" /> is its order type, then one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018067.png" />. Thus, 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018068.png" />. Different indices correspond to different initial numbers. Each ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018069.png" /> is the index of some initial number. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018070.png" /> is a limit ordinal number, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018071.png" /> is a regular initial number.

An initial number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018072.png" /> is called weakly inaccessible if it is regular and its index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018073.png" /> is a limit number. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018074.png" /> is weakly inaccessible, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018075.png" /> is singular and, thus, is not weakly inaccessible. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018076.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018077.png" /> is weakly inaccessible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018078.png" />.

Weakly-inaccessible ordinal numbers allow a classification similar to the classification of inaccessible cardinal numbers (cf. [[Cardinal number|Cardinal number]]). 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|transfinite induction]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018081.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018082.png" /> is a limit number. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018083.png" /> is called a power of a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018085.png" /> is the base of the power and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018086.png" /> is the exponent of the power. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018088.png" />, one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018091.png" />. The limit of this sequence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018092.png" />, is the least critical number of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018093.png" />, i.e. the least ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018094.png" /> among those for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018095.png" />. Numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018096.png" /> for which this equality holds are called epsilon-ordinals.

Raising to a power can be used to represent ordinal numbers in a form resembling decimal representation of positive integers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018098.png" />, then there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o07018099.png" /> and sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180101.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180104.png" />. Formula (1) for the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180106.png" /> satisfying the conditions (2) is called the representation of the ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180107.png" /> in the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180108.png" />. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180109.png" /> are called the digits, and the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180110.png" /> are called 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180111.png" /> is used to define the natural sum and the natural product of ordinal numbers.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Cantor,   "Contributions to the founding of the theory of transfinite numbers" , Dover, reprint  (1952)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Hausdorff,   "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Kuratowski,   A. Mostowski,   "Set theory" , North-Holland  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Sierpiński,   "Cardinal and ordinal numbers" , PWN  (1958)</TD></TR></table>



====Comments====
The ordinal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180112.png" />, the least ordinal number to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180113.png" /> is cofinal, is called the cofinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180114.png" />.

The ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180115.png" /> and (by the [[Axiom of choice|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180116.png" /> set theory are consistent, they remain so after the addition of the axiom stating that all initials with limit-index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180117.png" /> are singular. Thus, the axioms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180118.png" />, if consistent, cannot prove that there are any weakly-inaccessible ordinal numbers other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070180/o070180119.png" />.

For countable ordinal numbers see also [[Descriptive set theory|Descriptive set theory]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Kuratowski,   "Introduction to set theory and topology" , Pergamon  (1972)  (Translated from Polish)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.J. Jech,   "Set theory" , Acad. Press  (1978)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Barwise (ed.) , ''Handbook of mathematical logic'' , North-Holland  (1977)  ((especially the article of D.A. Martin on Descriptive set theory))</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Levy,   "Basic set theory" , Springer  (1979)</TD></TR></table>