In
axiomatic set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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, ...
, 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. Th ...
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 the ...
function; the gimel function is used for studying the
continuum function In mathematics, the continuum function is \kappa\mapsto 2^\kappa, i.e. raising 2 to the power of κ using cardinal exponentiation. Given a cardinal number, it is the cardinality of the power set of a set of the given cardinality.
See also
*Co ...
and the
cardinal exponentiation
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. The ...
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 d ...
.
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 In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. (extending a result of Robert M. Solovay) showed via forcing that the only constraints on permissible values for 2''κ'' when ''κ'' is a regular cardina ...
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 PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more ap ...
.
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
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
(assuming consistency).
Under this hypothesis cardinal exponentiation is simplified, though not to the extent of the
continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
(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
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named af ...
*
Beth number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second H ...
References
*
*
*
Thomas Jech
Thomas J. Jech ( cs, Tomáš Jech, ; born January 29, 1944 in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years.
Life
He was educated at Charles University (his advisor was Petr Vopěnka) and from 2 ...
, ''Set Theory'', 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, {{ISBN, 3-540-44085-2.
Cardinal numbers