# Some questions about $0^{\sharp}$ and forcing over $L$

1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is well-known that in $L[G],$ $0^\sharp$ does not exist, hence by Jensen's covering lemma the pair $(L, L[G])$ satisfies the covering lemma; i.e any uncountable set $X \in L[G]$ of ordinals is covered by a set $Y\in L$ of the same cardinality.

Is there a direct proof of this fact without using Jensen's covering lemma and using the properties of $P$ and the fact that we are forcing over $L$?

2-Work in $L$. Let $P$ be the Easton support product of forcing notions $Add(\kappa^+, 1), \kappa$ a singular cardinal. Then $P$ is tame, preserves cardinals and the $GCH$. Is there any tame class forcing notion $Q$ such that forcing with $P\times Q$ over $L$ collapses all cardinals into $\omega$

Remark. It is possible to define a tame and cardinal preserving class forcing notion $P$ over $L$ whose product $P\times P$ collapses all cardinals into $\omega$.

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This might be basic, but can you give an example of a tame cardinal-preserving forcing whose square collapses everything to $\omega$? That sounds really cool! – Noah Schweber Jul 21 '13 at 12:51
Assume $V=L$ and suppose there are no inaccessible cardinals. By a result of Jensen, for each regular cardinal $\kappa,$ there exists a $\kappa-$closed $\kappa^+-$Souslin tree $T_{\kappa^+}$ such that forcing with $T_{\kappa^+}\times T_{\kappa^+}$ collapses $\kappa^+$ into $\kappa.$ Let $P$ be the Easton support product of such $T_{\kappa^+}$'s. Then $P$ is as required. Just note that forcing with $P\times P$ collapses all $\kappa^+, \kappa$ regular, and hence by results of Shelah it collapses all uncountable cardinals. – Mohammad Golshani Jul 22 '13 at 3:12