Limit Cardinal - The Notion of Inaccessibility and Large Cardinals

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 (resp. strong) limit cardinal κ the requirement 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 (resp. 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.