# Gimel function

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

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.

For regular cardinals , , and Easton's theorem says we don't know much about the values of this function. For singular , upper bounds for 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
- 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 ℵ
_{0}≤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

## References

**Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).****Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).**- Thomas Jech,
*Set Theory*, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.