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]

  • Bukovský, L. (1965), "The continuum problem and powers of alephs", Comment. Math. Univ. Carolinae, 6: 181–197, MR 0183649
  • Jech, Thomas J. (1973), "Properties of the gimel function and a classification of singular cardinals" (PDF), Fund. Math., Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, I., 81 (1): 57–64, doi:10.4064/fm-81-1-57-64, MR 0389593
  • Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.