Constructive proof

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.

Values of the Gimel function

The gimel function has the property $\gimel (\kappa )>\kappa$ for all infinite cardinals κ by König's theorem.

For regular cardinals $\kappa$, $\gimel (\kappa )=2^{\kappa }$, and Easton's theorem says we don't know much about the values of this function. For singular $\kappa$, upper bounds for $\gimel (\kappa )$ can be found from Shelah's PCF theory.

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 $2^{\kappa }=\gimel (\kappa )$
• 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 κ^λ = 2^λ
• If μ^λ≥κ for some μ<κ then κ^λ = μ^λ
• If κ> λ and μ^λ<κ for all μ<κ and cf(κ)≤λ then κ^λ = κ^cf(κ)
• If κ> λ and μ^λ<κ for all μ<κ and cf(κ)>λ then κ^λ = κ

References

• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.