# Gimel function

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

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

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

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

## Values of the Gimel function

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

For regular cardinals $\kappa$, $\gimel(\kappa)= 2^\kappa$, and Easton's theorem says we don't know much about the values of this function. For singular $\kappa$, upper bounds for $\gimel(\kappa)$ 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 $2^\kappa = \gimel(\kappa)$
• If κ is infinite and singular and the continuum function is eventually constant below κ then $2^\kappa=2^{<\kappa}$
• If κ is a limit and the continuum function is not eventually constant below κ then $2^\kappa=\gimel(2^{<\kappa})$

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

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• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.