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 of all positive integers ordered by the relation is . The ordinal number of the set consisting of and of the numbers of the form , for ordered by the relation is . One says that an ordinal number is equal to (less than) an ordinal number , () if a set of type is similar to (a proper segment of) a set of type . For arbitrary ordinal numbers and one and only one of the following possibilities holds: either , or , or . The set of all ordinal numbers less than is well-ordered with type by the relation . Moreover, any set of ordinal numbers is well-ordered by the relation , i.e. any non-empty set of ordinal numbers contains a least ordinal number. For any set of ordinal numbers there exists an ordinal number greater than any ordinal number from . Accordingly, the set of all ordinal numbers does not exist. The smallest of the ordinal numbers following an ordinal number is called the successor of and is denoted by . The ordinal number is called the predecessor of the ordinal number . An ordinal number is called a limit (ordinal) number if it does not have a predecessor. Thus, is a limit number. Any ordinal number can be represented in the form , where is a limit number and is an integer, the sum is understood in the sense of addition of order types (cf. Order type).

A transfinite sequence of type , or an -sequence, is a function defined on . If the values of this sequence are ordinal numbers and implies that , then it is called an ascending sequence. Let denote a -sequence, where is a limit number. The least of the ordinal numbers greater than any , where , is called the limit of the sequence for and is denoted by . For instance, . An ordinal number is cofinal to a limit number if is the limit of an ascending -sequence: . The ordinal number is the least ordinal number to which 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 if it is the least among the ordinal numbers of cardinality (i.e. among the order types of well-ordered sets of cardinality ). Hence is the least initial number. The initial ordinal number of power is denoted by . The set of all initial ordinal numbers of infinite cardinality less than is well-ordered. If the ordinal number is its order type, then one puts . 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, . Different indices correspond to different initial numbers. Each ordinal number is the index of some initial number. If is a limit ordinal number, then is a regular initial number.

An initial number is called weakly inaccessible if it is regular and its index is a limit number. For instance, is weakly inaccessible, but is singular and, thus, is not weakly inaccessible. If , then is weakly inaccessible if and only if .

Weakly-inaccessible ordinal numbers allow a classification similar to the classification of inaccessible cardinal numbers (cf. 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: , , , where is a limit number. The number is called a power of a number , is the base of the power and is the exponent of the power. For example, if , , one obtains , , . The limit of this sequence, , is the least critical number of the function , i.e. the least ordinal number among those for which . Numbers 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 , , then there exists a positive integer and sequences and such that

(1)

(2)

for . Formula (1) for the numbers and satisfying the conditions (2) is called the representation of the ordinal number in the base . The numbers are called the digits, and the numbers 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 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 , the least ordinal number to which is cofinal, is called the cofinality of .

The ordinal number 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 set theory are consistent, they remain so after the addition of the axiom stating that all initials with limit-index are singular. Thus, the axioms of , if consistent, cannot prove that there are any weakly-inaccessible ordinal numbers other than .

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)