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.
is a serif form of the Hebrew letter gimel.
The gimel function has the property
, and Easton's theorem says we don't know much about the values of this function.
can be found from Shelah's PCF theory.
The gimel hypothesis states that
ℷ ( κ ) = max (
is the smallest value allowed by the axioms of Zermelo–Fraenkel set theory (assuming consistency).
Under this hypothesis cardinal exponentiation is simplified, though not to the extent of the continuum hypothesis (which implies the gimel hypothesis).
Bukovský (1965) showed that all cardinal exponentiation is determined (recursively) by the gimel function as follows.
The remaining rules hold whenever