Gimel function

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

:$gimelcolonkappamapstokappa^\{mathrm\{cf\}(kappa)\}$

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
*If κ is a limit and the continuum function is not eventually constant below κ then
*If ℵ₀≤κ≤λ then κ^λ = 2^λ
*If &mu;^&lambda;&ge;&kappa; for some &mu;<&kappa; then &kappa;^&lambda; = &mu;^&lambda;
*If &kappa;> &lambda; and &mu;^&lambda;<&kappa; for all &mu;<&kappa; and cf(&kappa;)>&lambda; then &kappa;^&lambda; = &kappa;
*If &kappa;> &lambda; and &mu;^&lambda;<&kappa; for all &mu;<&kappa; and cf(&kappa;)&le;&lambda; then &kappa;^&lambda; = &kappa;^cf(&kappa;)

References

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