In
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 conce ...
, Easton's theorem is a result on the possible
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. T ...
s 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 cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinite ...
are
:
(where cf(''α'') is 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 ...
of ''α'') and
:
Statement
If ''G'' is a
class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjuga ...
whose domain consists of
ordinals and whose range consists of ordinals such that
# ''G'' is non-decreasing,
# 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 ...
of
is greater than
for each ''α'' in the domain of ''G'', and
#
is regular for each ''α'' in the domain of ''G'',
then there is a model of ZFC such that
:
for each
in the domain of ''G''.
The proof of Easton's theorem uses
forcing with a
proper class of forcing conditions over a model satisfying the generalized continuum hypothesis.
The first two conditions in the theorem are necessary. Condition 1 is a well known property of cardinality, while condition 2 follows from
König's theorem.
In Easton's model the powersets of
singular cardinal
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, s ...
s have the smallest possible cardinality compatible with the conditions that 2
κ has cofinality greater than κ and is a non-decreasing function of κ.
No extension to singular cardinals
proved that a singular cardinal of uncountable cofinality cannot be the smallest cardinal for which the
generalized 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 ...
fails. This shows that Easton's theorem cannot be extended to the class of all cardinals. The program of
PCF theory gives results on the possible values of
for singular cardinals
. PCF theory shows that the values of the
continuum function on singular cardinals are strongly influenced by the values on smaller cardinals, whereas Easton's theorem shows that the values of the continuum function on
regular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinite ...
s are only weakly influenced by the values on smaller cardinals.
See also
*
Singular cardinal hypothesis
In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal.
According to Mitchell (1992), the sin ...
*
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 a ...
*
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
*
*{{citation, mr=0429564, authorlink=Jack Silver, last= Silver, first= Jack, chapter=On the singular cardinals problem, title= Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), volume= 1, pages= 265–268, publisher= Canad. Math. Congress, publication-place= Montreal, Que., year= 1975
Set theory
Theorems in the foundations of mathematics
Cardinal numbers
Forcing (mathematics)
Independence results