Sacks forcing
In mathematics, forcing is a method of constructing new models M[G] of set theory by adding a generic subset G of a poset P to a model M. The poset P used will determine what statements hold in the new universe (the 'extension'); to force a statement of interest thus requires construction of a suitable P. This article lists some of the posets P that have been used in this construction.
Amoeba forcing is forcing with the amoeba order, and adds a measure 1 set of random reals.
In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω2 × ω to {0,1} and p < q if p q.
This poset satisfies the countable chain condition. Forcing with this poset adds ω2 distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum hypothesis.
More generally, one can replace ω2 by any cardinal κ so construct a model where the continuum has size at least κ. Here, the only restriction is that κ does not have cofinality ω.
Grigorieff forcing (after Serge Grigorieff) destroys a free ultrafilter on ω.
Hechler forcing (after Stephen Herman Hechler) is used to show that Martin's axiom implies that every family of less than c functions from ω to ω is eventually dominated by some such function.
P is the set of pairs (s,E) where s is a finite sequence of natural numbers (considered as functions from a finite ordinal to ω) and E is a finite subset of some fixed set G of functions from ω to ω. The element (s, E) is stronger than (t,F) if t is contained in s, F is contained in E, and if k is in the domain of s but not of t then s(k)>h(k) for all h in F.
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