Class of cardinal numbers
In mathematics, limit cardinals are certain cardinal numbers. 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, formula_1 (aleph-naught), is a strong limit cardinal, and hence also a weak limit cardinal.
Constructions.
One way to construct limit cardinals is via the union operation: formula_2 is a weak limit cardinal, defined as the union of all the alephs before it; and in general formula_3 for any limit ordinal "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
formula_4
formula_5 (the smallest ordinal equinumerous with the powerset)
If "λ" is a limit ordinal, formula_6
The cardinal
formula_7
is a strong limit cardinal of cofinality ω. More generally, given any ordinal "α", the cardinal
formula_8
is a strong limit cardinal. Thus there are arbitrarily large strong limit cardinals.
Relationship with ordinal subscripts.
If the axiom of choice holds, every cardinal number has an initial ordinal. If that initial ordinal is formula_9 then the cardinal number is of the form formula_10 for the same ordinal subscript "λ". The ordinal "λ" determines whether formula_10 is a weak limit cardinal. Because formula_12 if "λ" is a successor ordinal then formula_10 is not a weak limit. Conversely, if a cardinal "κ" is a successor cardinal, say formula_14 then formula_15 Thus, in general, formula_10 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 formula_17 is a weak limit cardinal, but neither proves nor disproves that formula_17 is a strong limit cardinal (Hrbacek and Jech 1999:168). The generalized continuum hypothesis states that formula_19 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. 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.
formula_1 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 formula_1, due to Gödel's incompleteness theorem. More specifically, if formula_22 is weakly inaccessible then formula_23. These form the first in a hierarchy of large cardinals.