Forcing (mathematics) - Easton Forcing

Easton Forcing

The exact value of the continuum in the above Cohen model, and variants like Fin(ω × κ, 2) for cardinals κ in general, was worked out by Robert M. Solovay, who also worked out how to violate GCH (the generalized continuum hypothesis), for regular cardinals only, a finite number of times. For example, in the above Cohen model, if CH holds in V, then 2ℵ₀ = ℵ₂ holds in V.

W. B. Easton worked out the infinite and proper class version of violating the GCH for regular cardinals, basically showing the known restrictions (monotonicity, Cantor's theorem, and König's theorem) were the only ZFC provable restrictions. See 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 will fail to give a model of ZFC. For example, Fin ( ω × On, 2 ), where "On" is the proper class of all ordinals, will make the continuum a proper class. Fin ( ω, On ) will introduce a countable enumeration of the ordinals. In both cases, the resulting V is visibly not a model of ZFC.

At one time, it was thought that more sophisticated forcing would also allow arbitrary variation in the powers of singular cardinals. But this has turned out to be a difficult, subtle and even surprising problem, with several more restrictions provable in ZFC, and with the forcing models depending on the consistency of various large cardinal properties. Many open problems remain.