12
$\begingroup$

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 $\omega$.

$\endgroup$
2
  • 4
    $\begingroup$ 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! $\endgroup$ Jul 21, 2013 at 12:51
  • 3
    $\begingroup$ 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. $\endgroup$ Jul 22, 2013 at 3:12

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.