Effective forcing versus proper forcing Original Research Article

℘, a notion of forcing over E(ω1), the E-closure of L(ω1), is said to be effective if every sideways ℘-generic extension preserves E-closure. There are set notions of forcing in E(ω1) that do not preserve E-closure. The main theorem below asserts that ℘ is effective if and only if it is locally proper, a weak variant of Shelahʹs notion of proper.