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 that
Values of the Gimel function
The gimel function has the property for all infinite cardinals κ by König's theorem.
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 κ is an infinite regular cardinal (in particular any infinite successor) 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(κ) ≤ λ then κλ = κcf(κ)
- If κ > λ and μλ < κ for all μ < κ and cf(κ) > λ then κλ = κ
- Bukovský, L. (1965), "The continuum problem and powers of alephs", Comment. Math. Univ. Carolinae, 6: 181–197, MR 0183649
- Jech, Thomas J. (1973), "Properties of the gimel function and a classification of singular cardinals" (PDF), Fund. Math., Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, I., 81 (1): 57–64, doi:10.4064/fm-81-1-57-64, MR 0389593
- Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.