Abstract

It is proved (Theorem 1) that if 0♯ exists, then any constructible forcing property which over L adds no reals, over V collapses an uncountable L-cardinal to cardinality ω. This improves a theorem of Foreman, Magidor, and Shelah. Also, a method for approximating this phenomenon generically is found (Theorem 2). The strategy is first to reduce the problem of `disabling' forcing properties to that of specializing certain trees in a weak sense