Online research in philosophy

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.

It is shown that if κ is an uncountable successor cardinal in L[ 0 ♯ ], then there is a normal tree T ∈ L [ 0 ♯ ] of height κ such that $0^\sharp \not\in L\lbrack\mathbf{T}\rbrack$ . Yet T is $ -distributive in L[ 0 ♯ ]. A proper class version of this theorem yields an analogous L[ 0 ♯ ]-definable tree such that distinct branches in the presence of 0 ♯ collapse the universe. A heretofore unutilized method for (...) constructing in L[ 0 ♯ ] generic objects for certain L-definable forcings and "exotic sequences", combinatorial principles introduced by C. Gray, are used in constructing these trees. (shrink)