Shelah's strong covering property and CH in \(V[r]\) (Q2888627)

The authors provide a proof of a result of \textit{M. B. Vanliere} (see [Splitting the reals into two small pieces. Berkley: University of California (PhD thesis) (1982)]) which states that if \(X\subseteq \omega_n\) for some \(n\in\omega\), \(a\subseteq \omega\), \(L[X]\vDash \text{ZFC}+\text{GCH}\) and the cardinals of \(L[X]\) are the true cardinals, then \(L[X,a]\vDash \text{GCH}\). They examine Shelah's strong covering property in the context of pairs of models \((W,V)\), where \(V=W[r]\) for some real \(r\), and discuss various applications. For example, they show that if \(W\) and \(W[r]\) have the same cardinals, \(W\vDash \text{GCH}\) and \(W[r]\vDash 2^{\aleph_0} >\aleph_1\), then in \(W[r]\) there is an inner model with a measurable cardinal. In [\textit{S. Shelah} and \textit{H. Woodin}, J. Symb. Log. 49, 1185--1189 (1984; Zbl 0591.03028)] it is shown that there is a generic extension \(W\) of the constructible universe and reals \(a\), \(b\) such that both \(W[a]\), \(W[b]\) satisfy \(\text{CH}\), while in \(W[a,b]\) the continuum hypothesis fails. NEWLINENEWLINENEWLINE NEWLINEThe present paper gives a generalization of this result stating that, if \(\lambda\) is a cardinal of uncountable cofinality, there are \(\lambda\)-many measurable cardinals and \(\text{GCH}\) holds, then there is a cardinal-preserving generic extension \(W\) of the universe and reals \(a\), \(b\) such that \(W\), \(W[a]\), \(W[b]\) and \(W[a,b]\) have the same cardinals, \(W[a]\) and \(W[b]\) satisfy \(\text{GCH}\), while in \(W[a,b]\) the continuum has size \(\lambda\). The authors conclude with a discussion of open problems.