- Gimel function
In
axiomatic set theory , the gimel function is the following function mappingcardinal number s to cardinal numbers::gimelcolonkappamapstokappa^{mathrm{cf}(kappa)}
where cf denotes the
cofinality function; the gimel function is used for studying thecontinuum function and the cardinal exponentiation function.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'sPCF theory .Reducing the exponentiation function to the gimel function
All cardinal exponentiation is determined (recursively) by the gimel function as follows.
*If κ is an infinite successor cardinal then 2^kappa = gimel(kappa)
*If κ is a limit and the continuum function is eventually constant below κ then 2^kappa=2^{
*If κ is a limit and the continuum function is not eventually constant below κ then 2^kappa=gimel(2^{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 = κ
*If κ> λ and μλ<κ for all μ<κ and cf(κ)≤λ then κλ = κcf(κ)References
Thomas Jech , "Set Theory", 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.
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