We introduce the idea of a coherent adequate set of models, which can be used
as side conditions in forcing. As an application we define a forcing poset
which adds a square sequence on $\omega_2$ using finite conditions

Krueger, John

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John Krueger

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Fundamenta Mathematicae

Coherent adequate sets and forcing square

Abstract. We introduce the idea of a coherent adequate set of models, which can be used as side conditions in forcing. As an application we define a forcing poset which adds a square sequence on ω2 using finite conditions. In previous work [3] we introduced the idea of an adequate set of models and showed how to use adequate sets as side conditions in forcing with finite conditions. We gave several examples of forcing with adequate sets, including forcing posets for adding a generic function on ω2, adding a nonreflecting stationary subset of ω2, adding a Kurepa tree on ω1, and in [4] adding a club to a fat stationary subset of ω2. The main result of the present paper is to define a forcing poset using adequate sets which adds a ω1-sequence. The idea of using models as side conditions in forcing goes back to Todorčević [6], where the method was applied to add generic objects of size ω1 with finite approximations. In the original context of applications of PFA, the preservation of ω2 was not necessary. To preserve ω2, Todorčevic ́ introduced the requirement of a system of isomorphisms on the models in a condition

Coherent Adequate Sets and Forcing Square

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