Gimel Function - Reducing The Exponentiation Function To The Gimel Function

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