In
mathematics, limit cardinals are certain
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
s. A cardinal number ''λ'' is a weak limit cardinal if ''λ'' is neither a
successor cardinal nor zero. This means that one cannot "reach" ''λ'' from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.
A cardinal ''λ'' is a strong limit cardinal if ''λ'' cannot be reached by repeated
powerset operations. This means that ''λ'' is nonzero and, for all ''κ'' < ''λ'', 2
''κ'' < ''λ''. Every strong limit cardinal is also a weak limit cardinal, because ''κ''
+ ≤ 2
''κ'' for every cardinal ''κ'', where ''κ''
+ denotes the successor cardinal of ''κ''.
The first infinite cardinal,
(
aleph-naught
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named ...
), is a strong limit cardinal, and hence also a weak limit cardinal.
Constructions
One way to construct limit cardinals is via the union operation:
is a weak limit cardinal, defined as the union of all the alephs before it; and in general
for any
limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
''λ'' is a weak limit cardinal.
The
ב operation can be used to obtain strong limit cardinals. This operation is a map from ordinals to cardinals defined as
:
:
(the smallest ordinal
equinumerous
In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', the ...
with the powerset)
:If ''λ'' is a limit ordinal,
The cardinal
:
is a strong limit cardinal of
cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses the ...
ω. More generally, given any ordinal ''α'', the cardinal
:
is a strong limit cardinal. Thus there are arbitrarily large strong limit cardinals.
Relationship with ordinal subscripts
If the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
holds, every cardinal number has an
initial ordinal
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
. If that initial ordinal is
then the cardinal number is of the form
for the same ordinal subscript ''λ''. The ordinal ''λ'' determines whether
is a weak limit cardinal. Because
if ''λ'' is a successor ordinal then
is not a weak limit. Conversely, if a cardinal ''κ'' is a successor cardinal, say
then
Thus, in general,
is a weak limit cardinal if and only if ''λ'' is zero or a limit ordinal.
Although the ordinal subscript tells us whether a cardinal is a weak limit, it does not tell us whether a cardinal is a strong limit. For example,
ZFC proves that
is a weak limit cardinal, but neither proves nor disproves that
is a strong limit cardinal (Hrbacek and Jech 1999:168). The
generalized continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
states that
for every infinite cardinal ''κ''. Under this hypothesis, the notions of weak and strong limit cardinals coincide.
The notion of inaccessibility and large cardinals
The preceding defines a notion of "inaccessibility": we are dealing with cases where it is no longer enough to do finitely many iterations of the successor and powerset operations; hence the phrase "cannot be reached" in both of the intuitive definitions above. But the "union operation" always provides another way of "accessing" these cardinals (and indeed, such is the case of limit ordinals as well). Stronger notions of inaccessibility can be defined using
cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses the ...
. For a weak (respectively strong) limit cardinal ''κ'' the requirement is that cf(''κ'') = ''κ'' (i.e. ''κ'' be
regular) so that ''κ'' cannot be expressed as a sum (union) of fewer than ''κ'' smaller cardinals. Such a cardinal is called a
weakly (respectively strongly) inaccessible cardinal. The preceding examples both are singular cardinals of cofinality ω and hence they are not inaccessible.
would be an inaccessible cardinal of both "strengths" except that the definition of inaccessible requires that they be uncountable. Standard Zermelo–Fraenkel set theory with the axiom of choice (ZFC) cannot even prove the consistency of the existence of an inaccessible cardinal of either kind above
, due to
Gödel's incompleteness theorem. More specifically, if
is weakly inaccessible then
. These form the first in a hierarchy of
large cardinals.
See also
*
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
References
*
*
* {{Citation , last1=Kunen , first1=Kenneth , author1-link=Kenneth Kunen , title=Set theory: An introduction to independence proofs , publisher=
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as '' The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', ...
, isbn=978-0-444-86839-8 , year=1980
External links
* http://www.ii.com/math/cardinals/ Infinite ink on cardinals
Set theory
Cardinal numbers