0$^{\sharp}$ can be invisibly class generic.

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 there is no satisfactory first-order characterization of those subsets of ω 2 that have closed unbounded subsets in ω 1 , ω 2 and GCH preserving outer models. These “anticharacterization” results generalize to subsets of successors of uncountable regular cardinals. Similar results are proved for trees of height and cardinality κ + and for partitions of [ κ + ] 2 , when κ is an infinite cardinal.

Stanley, M.C., A Π12 singleton incompatible with 0#, Annals of Pure and Applied Logic 66 27–88. A non-constructible Π12 singleton that is absolute for ω-models of ZF is produced by class forcing over the minimum model.

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)