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Concept
Limit cardinal
Summary
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.
One way to construct limit cardinals is via the union operation: is a weak limit cardinal, defined as the union of all the alephs before it; and in general 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
(the smallest ordinal equinumerous with the powerset)
If λ is a limit ordinal,
The cardinal
is a strong limit cardinal of cofinality ω. More generally, given any ordinal α, the cardinal
is a strong limit cardinal. Thus there are arbitrarily large strong limit cardinals.
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).
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