Quotients of strongly proper forcings and guessing models (Q2805037)

The notion of a strongly proper forcing notion was introduced by \textit{W. J. Mitchell} [Trans. Am. Math. Soc. 361, No. 2, 561--601 (2009; Zbl 1179.03048)], and since then it found to be very useful in many forcing arguments. Recall that for a model \(M\) and a forcing notion \(\mathbb{P}\), a condition \(p \in \mathbb{P}\) is called \((M, \mathbb{P})\)-strongly generic if for all \(q \leq p\) there is a condition \(q |M \in M\) such that any \(r \leq q|M\) with \(r \in M\) is compatible with \(q\). The forcing notion \(\mathbb{P}\) is called \(\mathcal{S}\)-strongly proper, where \(\mathcal{S}\) is a collection of models, if for any \(M \in \mathcal{S}\) and any \(p \in M \cap \mathbb{P}\), there exists a condition \(q \leq p\) that is \((M, \mathbb{P})\)-strongly generic.NEWLINENEWLINEIn the paper under review, the authors define variants of strong genericity, in particular they define two strengthenings of strong genericity, namely simple and universal and discuss their properties. In particular, they investigate quotients of simple and universal strongly proper forcing notions, and show that under some circumstances, quotients of forcing posets which have such conditions for stationarily many \(N\) are well behaved.NEWLINENEWLINERecall that the principle ISP(\(\kappa^+\)) asserts that, for every sufficiently large regular cardinal \(\theta\), the set \(\{ M \prec H(\theta): |M| =\kappa, M \cap \kappa^+ \in \kappa^+\) and \((\bar{M}, V)\) satisfies the \(\kappa\)-approximation property\(\}\) is stationary in \(P_{\kappa^+}(H_\theta)\), where \(\bar{M}\) denotes the transitive collapse of \(M\).NEWLINENEWLINE\textit{M. Viale} and \textit{C. Weiß} [Adv. Math. 228, No. 5, 2672--2687 (2011; Zbl 1251.03059)] showed that ISP(\(\omega_2\)) implies \(2^{\aleph_0} \geq \aleph_2\) and they asked if it decides the value of the continuum. Using the method of adequate set forcing, developed by the second author [``Forcing with adequate sets of models as side conditions'' (submitted)], and using their previous results, the authors answer the question, by showing that ISP(\(\omega_2\)) is consistent with \(2^{\aleph_0}\) being arbitrary large.