In axiomatic set theory, the gimel function is the following function mapping cardinal number

Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...

s to cardinal numbers:


where cf denotes the cofinality

Cofinality

In mathematics, especially in order theory, the cofinality cf of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A....

 function; the gimel function is used for studying the continuum function

Continuum function

The continuum function is \kappa\mapsto 2^\kappa, i.e. raising 2 to the power of κ using cardinal exponentiation. Given a cardinal number, it is the cardinality of the power set of a set of the given cardinality.-See also:*Continuum hypothesis...

 and the cardinal exponentiation function.

Values of the Gimel function

The gimel function has the property for all infinite cardinals κ by König's theorem

König's theorem (set theory)

In set theory, König's theorem colloquially states that if the axiom of choice holds, I is a set, mi and ni are cardinal numbers for every i in I, and m_i In set theory, König's theorem In set theory, König's theorem (named after the Hungarian mathematician Gyula Kőnig, who published under the...

.

For regular cardinals
,
, and Easton's theorem

Easton's theorem

In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. showed via forcing that...

 says we don't know much about the values of this function. For singular
, upper bounds for can be found from Shelah

Saharon Shelah

Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.-Biography:...

's PCF theory

PCF theory

PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more applications as well...

.

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
• If κ is a limit 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 κ^λ = 2^λ
• If μ^λ≥κ for some μ<κ then κ^λ = μ^λ
• If κ> λ and μ^λ<κ for all μ<κ and cf(κ)≤λ then κ^λ = κ^cf(κ)
• If κ> λ and μ^λ<κ for all μ<κ and cf(κ)>λ then κ^λ = κ