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 is a serif form of the Hebrew letter gimel.

Values of the gimel function

by König's theorem.

, and Easton's theorem can be found from Shelah's PCF theory.

The gimel hypothesis

The gimel hypothesis 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 generalized 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.

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, .