Editing Forcing (mathematics) (section)

== Easton forcing ==
The exact value of the continuum in the above Cohen model, and variants like <math> \operatorname{Fin}(\omega \times \kappa,2) </math> for cardinals <math> \kappa </math> in general, was worked out by [[Robert M. Solovay]], who also worked out how to violate <math> \mathsf{GCH} </math> (the [[Continuum hypothesis#Generalized continuum hypothesis|generalized continuum hypothesis]]), for [[regular cardinal]]s only, a finite number of times. For example, in the above Cohen model, if <math> \mathsf{CH} </math> holds in <math> V </math>, then <math> 2^{\aleph_{0}} = \aleph_{2} </math> holds in <math> V[G] </math>.

[[William Bigelow Easton|William B. Easton]] worked out the proper class version of violating the <math> \mathsf{GCH} </math> for regular cardinals, basically showing that the known restrictions, (monotonicity, [[Cantor's theorem|Cantor's Theorem]] and [[König's theorem (set theory)|König's Theorem]]), were the only <math> \mathsf{ZFC} </math>-provable restrictions (see [[Easton's theorem|Easton's Theorem]]).

Easton's work was notable in that it involved forcing with a proper class of conditions. In general, the method of forcing with a proper class of conditions fails to give a model of <math> \mathsf{ZFC} </math>. For example, forcing with <math> \operatorname{Fin}(\omega \times \mathbf{On},2) </math>, where <math> \mathbf{On} </math> is the proper class of all ordinals, makes the continuum a proper class. On the other hand, forcing with <math> \operatorname{Fin}(\omega,\mathbf{On}) </math> introduces a countable enumeration of the ordinals. In both cases, the resulting <math> V[G] </math> is visibly not a model of <math> \mathsf{ZFC} </math>.

At one time, it was thought that more sophisticated forcing would also allow an arbitrary variation in the powers of [[Regular cardinal|singular cardinal]]s. However, this has turned out to be a difficult, subtle and even surprising problem, with several more [[PCF theory|restrictions provable]] in <math> \mathsf{ZFC} </math> and with the forcing models depending on the consistency of various [[large cardinal|large-cardinal]] properties. Many open problems remain.