Abstract
Several situations are presented in which there is an ordinal γ such that \({\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot(X) \in T \}}\) is a stationary subset of \({[\gamma]^{\aleph_0}}\) for all stationary \({S, T\subseteq \omega_1}\). A natural strengthening of the existence of an ordinal γ for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of \({H_{\omega_2}}\) and the existence of sharps for all reals. Also, an optimal model separating Bounded Semiproper Forcing Axiom (BSPFA) and Bounded Martin’s Maximum (BMM) is produced and it is shown that a strong form of BMM involving only parameters from \({H_{\omega_2}}\) implies that every function from ω 1 into ω 1 is bounded on a club by a canonical function.
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Asperó, D. On a convenient property about \({[\gamma]^{\aleph_0}}\) . Arch. Math. Logic 48, 653–677 (2009). https://doi.org/10.1007/s00153-009-0142-6
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DOI: https://doi.org/10.1007/s00153-009-0142-6