Relationship With Ordinal Subscripts
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 whether a cardinal is a weak limit, it does not tell 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.
Read more about this topic: Limit Cardinal
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