Quotients of strongly Proper Forcings and Guessing Models

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the ω1-approximation property. We prove that the existence of stationarily many ω1-guessing models in Pω2 (H(θ)), for sufficiently large cardinals θ, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss [13]. Many consistency results in set theory involve factoring a forcing poset Q over a regular suborder P in a forcing extension by P, and applying properties of the quotient forcing Q/ĠP. We will be interested in the situation where Q has strongly generic conditions for elementary substructures, and we wish the quotient Q/ĠP to have similar properties. For example, the quotient Q/ĠP having the approximation property is useful for constructing models in which there is a failure of square principles or related properties. We introduce some variations of strongly generic conditions, including simple and universal conditions. Our main theorem regarding quotients is that if Q is a forcing poset with greatest lower bounds for which there are stationarily many countable elementary substructures which have universal strongly generic conditions, and P is a regular suborder of Q which relates in a nice way to Q, then P forces that Q/ĠP has the ω1-approximation property. Several variations of this theorem are given, as well as an example which shows that not all quotients of strongly proper forcings are well behaved. Previously Weiss introduced combinatorial principles which characterize supercompactness yet also make sense for successor cardinals ([13], [14]). Of particular interest to us is the principle ISP(ω2), which asserts the existence of stationarily many ω1-guessing models in Pω1(H(θ)), for sufficiently large regular cardinals θ. This principle follows from PFA and has some of same consequences, such as the failure of the approachability property on ω1. It follows that ISP(ω2) implies that 2 ≥ ω2. Viale and Weiss [13] asked whether this principle settles the value of the continuum. We solve this problem by showing that ISP(ω2) is consistent with 2 ω being arbitrarily large. The solution is an application of the quotient theorem described above and the second author’s method of adequate set forcing ([3]). Date: Submitted June 2014; revised June 2015. 2010 Mathematics Subject Classification: Primary 03E40; Secondary 03E35.