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单词 CardinalExponentiationUnderGCH
释义

cardinal exponentiation under GCH


Many results about cardinal exponentiationMathworldPlanetmath can neither be proved nor disproved in ZFC. If, however, we allow ourselves to use GCH in addition to ZFC, then we have the following theorem, which gives an essentially completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath description of the way cardinal exponentiation involving infiniteMathworldPlanetmath cardinals works.

Theorem.

Assume the Generalized Continuum Hypothesis holds.Let κ and λ be cardinals,at least one of which is infinite,and such that κ>0 and λ>1.Then

λκ={κ+,𝑖𝑓λκ+;λ+,𝑖𝑓cf(λ)κλ;λ,𝑖𝑓κ<cf(λ).

Here, cf(λ) is the cofinality of λ, and λ+ is the cardinal successor of λ.

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更新时间:2025/5/5 2:49:43