In
axiomatic set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, the gimel function is the following function mapping
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
s to cardinal numbers:
:
where cf denotes the
cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses t ...
function; the gimel function is used for studying the
continuum function and the
cardinal exponentiation function. The symbol
is a serif form of the Hebrew letter
gimel
Gimel is the third letter of the Semitic abjads, including Phoenician Gīml , Hebrew Gimel , Aramaic Gāmal , Syriac Gāmal , and Arabic (in alphabetical order; fifth in spelling order). Its sound value in the original Phoenician and in all ...
.
Values of the gimel function
The gimel function has the property
for all infinite cardinals
by
König's theorem.
For regular cardinals
,
, and
Easton's theorem says we don't know much about the values of this function. For singular
, upper bounds for
can be found from
Shelah Shelah may refer to:
* Shelah (son of Judah), a son of Judah according to the Bible
* Shelah (name), a Hebrew personal name
* Shlach, the 37th weekly Torah portion (parshah) in the annual Jewish cycle of Torah reading
* Salih, a prophet described ...
's
PCF theory.
The gimel hypothesis
The gimel hypothesis states that
. In essence, this means that
for singular
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
is an infinite regular cardinal (in particular any infinite successor) then
*If
is infinite and singular and the continuum function is eventually constant below
then
*If
is a limit and the continuum function is not eventually constant below
then
The remaining rules hold whenever
and
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, {{ISBN, 3-540-44085-2.
Cardinal numbers