# Gimel function

Jump to navigation Jump to search

In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers:

${\displaystyle \gimel \colon \kappa \mapsto \kappa ^{\mathrm {cf} (\kappa )}}$

where cf denotes the cofinality function; the gimel function is used for studying the continuum function and the cardinal exponentiation function. The symbol ${\displaystyle \gimel }$ is a serif form of the Hebrew letter gimel.

The gimel hypothesis states that ${\displaystyle \gimel (\kappa )=\max(2^{{\text{cf}}(\kappa )},\kappa ^{+})}$

## Values of the Gimel function

The gimel function has the property ${\displaystyle \gimel (\kappa )>\kappa }$ for all infinite cardinals κ by König's theorem.

For regular cardinals ${\displaystyle \kappa }$, ${\displaystyle \gimel (\kappa )=2^{\kappa }}$, and Easton's theorem says we don't know much about the values of this function. For singular ${\displaystyle \kappa }$, upper bounds for ${\displaystyle \gimel (\kappa )}$ can be found from Shelah's PCF theory.

## 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 ${\displaystyle 2^{\kappa }=\gimel (\kappa )}$
• If κ is infinite and singular and the continuum function is eventually constant below κ then ${\displaystyle 2^{\kappa }=2^{<\kappa }}$
• If κ is a limit and the continuum function is not eventually constant below κ then ${\displaystyle 2^{\kappa }=\gimel (2^{<\kappa })}$

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

## References

• 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.