# ⓘ 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
• 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
• and functioned as youth section, without a senior team. In 2014, the club was resurrected by local businessmen. The club is currently in Liga Gimel Tel