Easton's theorem

The set of Easton, named after William Bigelow Easton, is a mathematical theorem in the field of set theory. The generalized continuum hypothesis, which can be in the set theory ZFC, that is, in Zermelo -Fraenkel set theory with the axiom of choice, neither proved nor disproved, says that the cardinality of the power set of a cardinal number, always with the successor cardinal number of matches. To prove the unprovable Paul Cohen had a model is constructed in which this hypothesis is false. The set of Easton provides also state that the generalized continuum hypothesis can be violated for regular cardinal numbers in almost any way.

Wording of the sentence

It is the class of all cardinal numbers and the partial class of regular cardinals. Next is a function with the following properties:

• Is monotonic, ie for.
• The cofinality of is strictly greater than, that is all.

Then there is a ZFC - model for all.

Comments

The sentence was proved in 1970 by means of generalized Easton forcing methods.

The continuum function is trivially monotonic and fulfilled by a set of inference from the king also the inequality. That's all you can say about the continuum function at regular points in ZFC, because according to the above set of Easton there is any function that satisfies these two conditions for regular cardinals ZFC - models in which the continuum function is exactly this feature. In this sense, it can be almost any wrong the generalized continuum function.

The simple continuum hypothesis, in Aleph notation can be arbitrarily wrong. After the set of Easton there are any cardinal with uncountable cofinality ZFC - models, is considered in which. For example, the equations are relatively consistent.

By the theorem of Silver is the least cardinal for which the equation is violated, not a singular cardinal of uncountable cofinality with. Therefore, the set of Easton can not be extended to singular cardinals.