@MISC{Schindler08onpatterns, author = {Ralf-dieter Schindler and Foreman Have}, title = {On patterns of cardinals with the tree property}, year = {2008} }

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Abstract

shown in [2], starting from ω many supercompact cardinals, that consistently the following holds. (⋆) For every n < ω, 2 ℵn = ℵn+2 and ℵn+1 has the tree property. Recall that a cardinal κ is said to have the tree property if there is no Aronszajn κ-tree, i.e. if every tree of height κ all of whose levels have size < κ admits a cofinal branch. We here show (in a certain sense of ”show”): Theorem 0.1 Suppose that there are δ1 < δ2 < δ3 <... with supremum σ such that σ is a strong limit cardinal, and for all n < ω, δ2n+2 = (δ2n+1) + and δn+1 has the tree property. Let G be Col(ω, < σ)-generic over V, and let