Abstract. 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 ω 1 -approximation property. We prove that the existence of stationarily many ω 1 -guessing models in Pω 2 (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/Ġ P . We will be interested in the situation where Q has strongly generic conditions for elementary substructures, and we wish the quotient Q/Ġ P to have similar properties. For example, the quotient Q/Ġ P 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/Ġ P has the ω 1 -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(ω 2 ), which asserts the existence of stationarily many ω 1 -guessing models in P ω1 (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 ω 1 . It follows that ISP(ω 2 ) implies that 2 ω ≥ ω 2 . Viale and Weiss [13] asked whether this principle settles the value of the continuum. We solve this problem by showing that ISP(ω 2 ) 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]).

Given an uncountable regular cardinal κ, a partial order is κstationarily layered if the collection of regular suborders of P of cardinality less than κ is stationary in Pκ(P). We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal κ is weakly compact if and only if every partial order satisfying the κ-chain condition is κ-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all κ-Knaster partial orders are κ-stationarily layered implies that κ is a Mahlo cardinal and every stationary subset of κ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.2010 Mathematics Subject Classification. 03E05, 03E55, 06A07.

We prove that Martin's Maximum does not imply the Diagonal Reflection Principle for stationary subsets of [ω2] ω . IntroductionIn Foreman-Magidor-Shelah [5], it was shown that Martin's Maximum MM implies the following stationary reflection principle, which is called the Weak Reflection Principle:WRP ≡ For any cardinal λ ≥ ω 2 and any stationaryWRP is known to have many interesting cosequences such as Chang's Conjecture (Foreman-Magidor-Shelah [5]), the presaturation of the non-stationary ideal over ω 1 (Feng-Magidor [4]), 2 ω ≤ ω 2 (folklore) and the Singular Cardinal Hypothesis (Shelah [12]). As for stationary reflection principles, simultaneous reflection is often discussed. Larson [10] proved that MM also implies the following simultaneous reflection principle of ω 1 -many stationary sets:WRP ω1 ≡ For any cardinal λ ≥ ω 2 and any sequence X ξ | ξ < ω 1 of stationary subsets ofCox [2] formulated the following strengthening of WRP ω1 , which is called the Diagonal Reflection Principle:DRP ≡ For any cardinal λ ≥ ω 2 and any sequence X α | α < λ of stationary subsets ofRecently, Fuchino-Ottenbreit-Sakai [6] proved that a variation of DRP is equivalent to some variation of the downward Löwenheim-Skolem theorem of the stationary logic. Cox [2] also introduced the following weakning of DRP, where X ⊆ [λ] ω is said to be projectively stationary if the set {x ∈ X | x ∩ ω 1 ∈ S} is stationary in [λ] ω for any stationary S ⊆ ω 1 :wDRP ≡ For any cardinal λ ≥ ω 2 and any sequence X α | α < λ of projectively stationary subsets ofCox [2] proved that MM implies wDRP, but it remained open whether MM implies DRP. In this paper, we prove that MM does not imply DRP. In fact, we prove slightly more.To state our result, we recall +-versions of the forcing axiom. For a class Γ of forcing notions and a cardinal µ ≤ ω 1 , MA +µ (Γ) is the following statement: *

We prove a variety of theorems about stationary set reflection and concepts related to internal approachability. We prove that an implication of Fuchino–Usuba relating stationary reflection to a version of Strong Chang’s Conjecture cannot be reversed; strengthen and simplify some results of Krueger about forcing axioms and approachability; and prove that some other related results of Krueger are sharp. We also adapt some ideas of Woodin to simplify and unify many arguments in the literature involving preservation of forcing axioms.