ⓘ 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
                                     
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  • Gimel is the third letter of the Semitic abjads, including Phoenician Gīml Hebrew ˈGimel ג, Aramaic Gāmal Syriac Gāmal ܓ, and Arabic ǧīm ج in alphabetical
  • 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
  • the ancient Guanches of the Canary Islands, also used throwing batons. Gimel the third letter of many Semitic alphabets, may have been named after a
  • called a tagin when written in a Sefer Torah Torah scroll besides ayin, gimel teth, nun, shin, and tzadi. For the Biblical and Modern Hebrew meaning
  • letter for the g phoneme gīml and as such is cognate with Hebrew gimel ג. Based on its name, the letter has been interpreted as an abstract representation
  • could be used to depict both characters. gimel gimel symbol is not the same as U 05D2 Hebrew letter gimel although the same glyph could be used to
  • one of six letters that represents two associated sounds the others are Gimel Dalet, Kaph, Pe and Taw When Beth has a hard pronunciation qûssāya
  • one of the six letters which can receive a Dagesh Kal. The six are Bet, Gimel Daleth, Kaph, Pe, and Tav. A notable variation on the letter Pe is the
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