Gimel function explained

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

\gimel\colon\kappa\mapsto\kappa ^{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

\gimel is a serif form of the Hebrew letter gimel.

Values of the gimel function

The gimel function has the property

\gimel(\kappa)>\kappa for all infinite cardinals \kappa by König's theorem.

For regular cardinals

\kappa , \gimel(\kappa) = 2 ^{\kappa} , and Easton's theorem says we don't know much about the values of this function. For singular \kappa , upper bounds for \gimel(\kappa) can be found from Shelah's PCF theory.

The gimel hypothesis

The gimel hypothesis states that

\gimel(\kappa) =max ( 2 ^{cf(\kappa)} ,\kappa ^{+)} . In essence, this means that \gimel(\kappa) for singular \kappa is the smallest value allowed by the axioms of Zermelo–Fraenkel set theory (assuming consistency).

Under this hypothesis cardinal exponentiation is simplified, though not to the extent of the continuum hypothesis (which implies the gimel hypothesis).

Reducing the exponentiation function to the gimel function

showed that all cardinal exponentiation is determined (recursively) by the gimel function as follows.

• If

\kappa is an infinite regular cardinal (in particular any infinite successor) then 2 ^{\kappa} = \gimel(\kappa)

• If

\kappa is infinite and singular and the continuum function is eventually constant below \kappa then 2 ^{\kappa=2} ^{<\kappa}

• If

\kappa is a limit and the continuum function is not eventually constant below \kappa then 2 ^{\kappa=\gimel(2} ^{<\kappa} )

The remaining rules hold whenever

\kappa and λ are both infinite:

• If then
• If for some then
• If and for all and then
• If and for all and then

See also

• Aleph number
• Beth number

References

• • • Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, .