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Values of the Gimel function
The gimel function has the property for all infinite cardinals κ by König's theorem.
Reducing the exponentiation function to the gimel function
All cardinal exponentiation is determined (recursively) by the gimel function as follows.
- If κ is an infinite successor cardinal then
- If κ is a limit and the continuum function is eventually constant below κ then
- If κ is a limit and the continuum function is not eventually constant below κ then
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 = κcf(κ)
- If κ> λ and μλ<κ for all μ<κ and cf(κ)>λ then κλ = κ