ⓘ Gimel function. In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers: ℷ: κ ↦ κ c f κ {\displaystyl ..


ⓘ Gimel function

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

ℷ: κ ↦ κ c f κ {\displaystyle \gimel \colon \kappa \mapsto \kappa ^{\mathrm {cf} \kappa}}

where cf denotes the cofinality function; the gimel function is used for studying the continuum function and the cardinal exponentiation function. The symbol ℷ {\displaystyle \gimel } is a serif form of the Hebrew letter gimel.

The gimel hypothesis states that ℷ κ = max 2 cf κ, κ +) {\displaystyle \gimel \kappa=\max2^\kappa},\kappa ^{+})}


1. Values of the Gimel function

The gimel function has the property ℷ κ > κ {\displaystyle \gimel \kappa> \kappa } for all infinite cardinals κ by Konigs theorem.

For regular cardinals κ {\displaystyle \kappa }, ℷ κ = 2 κ {\displaystyle \gimel \kappa=2^{\kappa }}, and Eastons theorem says we dont know much about the values of this function. For singular κ {\displaystyle \kappa }, upper bounds for ℷ κ {\displaystyle \gimel \kappa} can be found from Shelahs PCF theory.


2. Reducing the exponentiation function to the gimel function

Bukovsky 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 {\displaystyle 2^{\kappa }=\gimel \kappa}
  • If κ is infinite and singular and the continuum function is eventually constant below κ then 2 κ = 2 < κ {\displaystyle 2^{\kappa }=2^{ λ and μ λ < κ for all μ < κ and cfκ > λ then κ λ = κ
  • Gimel may mean Gimel Switzerland, a municipality in the canton of Vaud Gimel - les - Cascades, France Gimel letter a Hebrew letter Yom Gimel or simply
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  • cardinality of the power set of a set of the given cardinality. Continuum hypothesis Cardinality of the continuum Beth number Easton s theorem Gimel function
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