The gimel function has the property for all infinite cardinals by
König's theorem.
For regular cardinals
,
, and
Easton's theorem says we don't know much about the values of this function. For singular
, upper bounds for can be found from
Shelah's
PCF theory.
The gimel hypothesis
The gimel hypothesis states that . In essence, this means that for singular 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).
Reducing the exponentiation function to the gimel function
Bukovský (1965) showed that all cardinal exponentiation is determined (recursively) by the gimel function as follows.
If is an infinite regular cardinal (in particular any infinite successor) then
If is infinite and singular and the continuum function is eventually constant below then
If is a limit and the continuum function is not eventually constant below then
The remaining rules hold whenever and 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