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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 mathema ...
, Easton's theorem is a result on the possible
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s of
powerset In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
s. (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 are : \kappa < \operatorname(2^\kappa) (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''. Formally, :\operatorname(A) = \inf \ This definition of cofinality relies o ...
of ''α'') and : \text \kappa < \lambda \text 2^\kappa\le 2^\lambda.


Statement

If G is a class function 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''. Formally, :\operatorname(A) = \inf \ This definition of cofinality relies o ...
of \aleph_ is greater than \aleph_\alpha for each \alpha in the domain of G , and # \aleph_\alpha is regular for each \alpha in the domain of G , then there is a model of ZFC such that :2^ = \aleph_ for each \alpha in the domain of G . The proof of Easton's theorem uses forcing with a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
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 or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singular ...
s have the smallest possible cardinality compatible with the conditions that 2^ has cofinality greater than \kappa and is a non-decreasing function of \kappa .


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, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
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 2^\lambda for singular cardinals \lambda. 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 cardinals are only weakly influenced by the values on smaller cardinals.


See also

* Singular cardinal hypothesis *
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. They were introduced by the mathematician Georg Cantor and are named after the symbol he used t ...
*
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 Hebrew lett ...


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