Gimel function

The gimel function has the property for all infinite cardinals by König's theorem.

For regular cardinals , , and Easton's theorem says that very little about this function can be determined in ZFC without additional axioms. For singular , upper bounds for can be found from Shelah's PCF theory.

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 generalized continuum hypothesis (which implies the gimel hypothesis).

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 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