Gimel function - Alchetron, The Free Social Encyclopedia

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

Values of the Gimel function

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

For regular cardinals κ , ℷ ( κ ) = 2 κ , and Easton's theorem says we don't know much about the values of this function. For singular κ , upper bounds for ℷ ( κ ) 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 κ = ℷ ( κ )

If κ is infinite and singular and the continuum function is eventually constant below κ then 2 κ = 2 < κ

If κ is a limit and the continuum function is not eventually constant below κ then 2 κ = ℷ ( 2 < κ )

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

References

Gimel function Wikipedia

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Contents

Values of the Gimel function

Reducing the exponentiation function to the gimel function

References