Gimel function

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In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers:

\gimel\colon\kappa\mapsto\kappa^{\mathrm{cf}(\kappa)}

where cf denotes the cofinality function; the gimel function is used for studying the continuum function and the cardinal exponentiation function. The gimel function has the property \gimel(\kappa)>\kappa for all infinite cardinals κ by König's theorem.

All cardinal exponentiation is determined (recursively) by the gimel function as follows.

  • If κ is an infinite successor cardinal then 2^\kappa = \gimel(\kappa)
  • If κ is a limit and the continuum function is eventually constant below κ then 2^\kappa=2^{<\kappa}\times\gimel(\kappa)
  • If κ is a limit and the continuum function is not eventually constant below κ then 2^\kappa=\gimel(2^{<\kappa})

The remaining rules hold whenever κ and λ are both infinite:

  • If ℵ0≤κ≤λ then κλ = 2λ
  • If μλ≥κ for some μ<κ then κλ = μλ
  • If κ> λ and μλ<κ for all μ<κ and cf(κ)>λ then κλ = κ
  • If κ> λ and μλ<κ for all μ<κ and cf(κ)≤λ then κλ = κcf(κ)
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