Abstract
We continue the study of adequate sets which we began in (Krueger in Forcing with adequate sets of models as side conditions) by introducing the idea of a strongly adequate set, which has an additional requirement on the overlap of two models past their comparison point. We present a forcing poset for adding a club to a fat stationary subset of ω 2 with finite conditions, thereby showing that a version of the forcing posets of Friedman (Set theory: Centre de Recerca Matemàtica, Barcelona, 2003–2004, Trends in Mathematics. Birkhäuser Verlag, 2006) and Mitchell (Trans Am Math Soc 361(2):561601, 2009) for adding a club on ω 2 can be developed in the context of adequate sets.
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References
Krueger, J.: Forcing with adequate sets of models as side conditions (to appear)
Friedman, S.D.: Forcing with finite conditions. In: Set Theory: Centre de Recerca Matemàtica, Barcelona, 2003–2004, Trends in Mathematics, pp. 285–295. Birkhäuser Verlag, Basel (2006)
Mitchell W.: I[ω 2] can be the nonstationary ideal on Cof(ω 1). Trans. Am. Math. Soc. 361(2), 561601 (2009)
Neeman, I.: Forcing with sequences of models of two types (to appear)
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Krueger, J. Strongly adequate sets and adding a club with finite conditions. Arch. Math. Logic 53, 119–136 (2014). https://doi.org/10.1007/s00153-013-0358-3
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DOI: https://doi.org/10.1007/s00153-013-0358-3