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. The first infinite cardinal, ℵ 0 {\displaystyle \aleph _{0}} (aleph-naught), is a strong limit cardinal, and hence also a weak limit cardinal.
Constructions One way to construct limit cardinals is via the union operation: ℵ ω {\displaystyle \aleph _{\omega }} is a weak limit cardinal, defined as the union of all the alephs before it; and in general ℵ β {\displaystyle \aleph _{\beta }} for any limit ordinal β 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
ℶ 0 = ℵ 0 , {\displaystyle \beth _{0}=\aleph _{0},}
ℶ α + 1 = 2 ℶ α , {\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }},} (the smallest ordinal equinumerous with the powerset) If β is a limit ordinal, ℶ β = ⋃ { ℶ α : α < β } . {\displaystyle \beth _{\beta }=\bigcup \{\beth _{\alpha }:\alpha <\beta \}.}
The cardinal
ℶ ω = ⋃ { ℶ 0 , ℶ 1 , ℶ 2 , … } = ⋃ n < ω ℶ n {\displaystyle \beth _{\omega }=\bigcup \{\beth _{0},\beth _{1},\beth _{2},\ldots \}=\bigcup _{n<\omega }\beth _{n}}
is a strong limit cardinal of cofinality ω. More generally, given any ordinal α, the cardinal
ℶ α + ω = ⋃ n < ω ℶ α + n {\displaystyle \beth _{\alpha +\omega }=\bigcup _{n<\omega }\beth _{\alpha +n}}
is a strong limit cardinal. Thus there are arbitrarily large strong limit cardinals.
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