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 concer ...

, 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

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 p ...

s. (extending a result of Robert M. Solovay) showed via

forcing Forcing may refer to: Mathematics and science * Forcing (mathematics), a technique for obtaining independence proofs for set theory *Forcing (recursion theory), a modification of Paul Cohen's original set theoretic technique of forcing to deal with ...

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''. This definition of cofinality relies on the axiom of choice, as it uses t ...

of ''α'') and :

\text \kappa < \lambda \text 2^\kappa\le 2^\lambda.

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 conjug ...

whose domain consists of ordinals and whose range consists of ordinals such that # ''G'' is non-decreasing, # the

\aleph_

is greater than

\aleph_\alpha

for each ''α'' in the domain of ''G'', and #

\aleph_\alpha

is regular for each ''α'' 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

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 cardinals 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 t ...

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.

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

Statement

No extension to singular cardinals

See also

References