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 is a serif form of the Hebrew letter gimel.
The gimel hypothesis states thatValues of the Gimel function
The gimel function has the property for all infinite cardinals κ 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.Reducing the exponentiation function to the gimel function
showed that all cardinal exponentiation is determined by the gimel function as follows.
- If κ is an infinite regular cardinal then
- If κ is infinite and singular 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
The remaining rules hold whenever κ and λ 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 ≤ λ then κλ = κcf
- If κ > λ and μλ < κ for all μ < κ and cf > λ then κλ = κ
