Gimel function

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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[edit]

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[edit]

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 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[edit]