Bounded dagger principles

摘要

For an uncountable cardinal κ, let (?)_κ be the assertion that every ω1-stationary preserving poset of size ≤ κ is semiproper. We prove that (?)_(ω2) is a strong principle which implies a strong form of Chang's conjecture. We also show that (?)_(2ω1) implies that NS_(ω1) is presaturated.