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 cardinal 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, 2κ < λ. Every strong limit cardinal is also a weak limit cardinal, because by Cantor's theorem κ+ ≤ 2κ for every cardinal κ, where κ+ denotes the successor cardinal of κ.

(aleph-naught), is a strong limit cardinal, and hence also a weak limit cardinal.

Constructions

for any limit ordinal 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

1. (the smallest ordinal equinumerous with the powerset)

2. If

The cardinal

is a strong limit cardinal of cofinality . More generally, given any ordinal , the cardinal

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 for the same ordinal subscript . The ordinal if is not a weak limit. Conversely, if a cardinal is a weak limit cardinal if and only if is zero or a limit ordinal.

Although the aleph 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 is a strong limit cardinal (Hrbacek and Jech 1999:168). The generalized continuum hypothesis for every infinite cardinal . Under this hypothesis, the notions of weak and strong limit cardinals coincide.

The notion of inaccessibility and large cardinals

The concepts of weak or strong limit cardinals define notions 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 are both singular cardinals of cofinality ω, and hence they are not inaccessible.

, due to Gödel's incompleteness theorem. Inaccessible cardinals form the first in a hierarchy of large cardinals.

References

External links

• http://www.ii.com/math/cardinals/ Infinite ink on cardinals