Шексіз кардиналдар класы: Жұп және күшті лимит кардиналдары

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, 2κ < λ. Every strong limit cardinal is also a weak limit cardinal, because κ+ ≤ 2κ for every cardinal κ, where κ+ denotes the successor cardinal of κ. The first infinite cardinal, (aleph naught), is a strong limit cardinal, and hence also a weak limit cardinal.

If the axiom of choice holds, every cardinal number has an initial ordinal. 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 states that for every infinite cardinal κ. Under this hypothesis, the notions of weak and strong limit cardinals coincide.

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. 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.