Difference between revisions of "Cardinal number"
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Revision as of 17:14, 6 December 2014
2020 Mathematics Subject Classification: Primary: 03E10 [MSN][ZBL]
transfinite number, power in the sense of Cantor, cardinality of a set
That property of the set which is intrinsic to any set with the same cardinality as
. In this connection, two sets
and
are said to have the same cardinality (or to be equivalent) if there is a one-to-one onto function
with domain of definition
and set of values
. G. Cantor defined the cardinal number of a set as that property of it which remains after abstracting the qualitative nature of its elements and their ordering. By way of stressing the double act of abstraction, Cantor introduced the symbol
to denote the cardinal number of
. The most commonly used from among the various notations for a cardinal number are the symbols
and
. If
is a finite set containing
elements, then
. If
denotes the set of natural numbers, then
(see Aleph). If
denotes the set of real numbers, then
, the power of the continuum. The set
of all subsets of
is not equivalent to
or to any subset of
(Cantor's theorem). In particular, no two of the sets
![]() | (1) |
are equivalent. When , the above sequence gives rise to infinitely many distinct infinite cardinal numbers. Further cardinal numbers are obtained by taking the union
of the sets in (1) and constructing the analogous sequence, setting
. This process can be continued infinitely often. The scale (class) of all infinite cardinal numbers is much richer than the scale (class) of finite cardinals. Furthermore, there are so many of them that it is not possible to form a set containing at least one of each cardinal number.
One can define for cardinal numbers the operations of addition, multiplication, raising to a power, as well as taking the logarithm and extracting a root. Thus, the cardinal number is the sum of
and
,
, if each set of cardinality
can be represented as a disjoint union of sets
and
of cardinalities
and
, respectively; the cardinal number
is the product of
and
,
, if
is the cardinal number of the Cartesian product
where
and
. Addition and multiplication of cardinal numbers is commutative and associative, and multiplication is distributive with respect to addition. The cardinal number
is the power with base
and exponent
,
, if every set of cardinality
is equivalent to the set
of all functions
, where
and
. The cardinal number
is said to be smaller or equal to the cardinal number
,
, if every set of cardinality
is equivalent to some subset of a set of cardinality
. If
and
, then
(the Cantor–Bernstein theorem), so that the scale of cardinal numbers is totally ordered. Furthermore, for each cardinal number
the set
is totally well-ordered, which enables one to define the logarithm
of
to the base
,
, as the smallest cardinal number
such that
; similarly, the
-th root
of the cardinal number
is the smallest cardinal number
such that
.
Any cardinal number can be identified with the smallest ordinal number of cardinality
. In particular,
corresponds to the ordinal number
,
to the ordinal
, etc. Thus, the scale of cardinal numbers is a subscale of the scale of ordinal numbers. A number of properties of ordinal numbers carry over to cardinal numbers; however, these same properties can also be defined "intrinsically" . If
for every
and if
, then
![]() | (2) |
(König's theorem). If in (2) one sets and
, then
![]() | (3) |
In particular, for any it is impossible to express the power
as the sum of an infinite increasing sequence of length
all terms of which are less than
. For each cardinal number
, the cofinal character of
, denoted by
, is defined as the smallest cardinal number
such that
can be written as
for suitable
and
. If
, then
is called regular, otherwise it is called singular. For each cardinal number
, the smallest cardinal number
greater than
is regular (granted the axiom of choice). An example of a singular cardinal number is the cardinal number
on the left-hand side of (3) under the condition that
. In this case
![]() |
A cardinal number is called a limit cardinal number if for any
there exists a
such that
. Examples of limit cardinal numbers are
and
, while
is a non-limit cardinal number. A regular limit cardinal number is called weakly inaccessible. A cardinal number
is said to be a strong limit cardinal if for any
,
. A strong regular limit cardinal number is called strongly inaccessible. It follows from the generalized continuum hypothesis that the classes of strongly- (or weakly-) inaccessible cardinal numbers coincide. The classes of inaccessible cardinal numbers can be further classified (the so-called Malo scheme), which leads to the definition of hyper-inaccessible cardinal numbers. The assertion that strongly- (or weakly-) inaccessible cardinal numbers exist happens to be independent of the usual axioms of axiomatic set theory.
A cardinal number is said to be measurable (more precisely,
-measurable), if there exists a set
of cardinality
and a function
defined on all elements of the set
, taking the values
or
and such that
,
for any
and such that if
is a sequence of pairwise disjoint subsets of
, then
![]() |
Every cardinal number less than the first uncountable strongly-inaccessible cardinal number is non-measurable (Ulam's theorem), and the first measurable cardinal number is certainly strongly inaccessible. However, the first measurable cardinal number is considerably larger than the first uncountable strongly-inaccessible cardinal number (Tarski's theorem). It is not known (1987) whether the hypothesis that measurable cardinal numbers exist contradicts the axioms of set theory.
References
[1] | P.S. [P.S. Aleksandrov] Alexandroff, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian) |
[2] | G. Cantor, , New ideas in mathematics , Handbook Math. Libraries , 6 (1914) pp. 90–184 (In Russian) |
[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 (1965) (Translated from Polish) |
Comments
König's theorem stated above is usually called the König–Zermelo theorem.
References
[a1] | T.J. Jech, "Set theory" , Acad. Press (1978) pp. Chapt. 7 (Translated from German) |
[a2] | A. Levy, "Basic set theory" , Springer (1979) |
Cardinal number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cardinal_number&oldid=35406