Gimel function

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

where cf denotes the cofinality function; the gimel function is used for studying the continuum function and the cardinal exponentiation function. The symbol $\gimel$ is a serif form of the Hebrew letter gimel.

The gimel hypothesis states that $\gimel (\kappa )=\max(2^{{\text{cf}}(\kappa )},\kappa ^{+})$

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.