The Harrington-shelah Model with Large Continuum

We prove from the existence of a Mahlo cardinal the consistency of the statement that 2ω = ω3 holds and every stationary subset of ω2∩cof(ω) reflects to an ordinal less than ω2 with cofinality ω1. Let us say that stationary set reflection holds at ω2 if for any stationary set S ⊆ ω2∩cof(ω), there is an ordinal α ∈ ω2∩cof(ω1) such that S∩α is stationary in α (that is, S reflects to α). In a classic forcing construction, Harrington and Shelah [2] proved the equiconsistency of stationary set reflection at ω2 with the existence of a Mahlo cardinal. Specifically, if stationary set reflection holds at ω2, then ω1 fails, and hence ω2 is a Mahlo cardinal in L. Conversely, if κ is a Mahlo cardinal, then the generic extension obtained by Lévy collapsing κ to become ω2 and then iterating to kill the stationarity of nonreflecting sets satisfies stationary set reflection at ω2. The Harrington-Shelah argument is notable because the majority of stationary set reflection principles are derived by extending large cardinal elementary embeddings, and thus use large cardinal principles much stronger than the existence of a Mahlo cardinal. The original Harrington-Shelah model satisfies the generalized continuum hypothesis, and in particular, that 2 = ω1. Suppose we would like to obtain a model of stationary set reflection at ω2 together with 2 ω = ω2. A natural construction would be to iterate forcing with countable support of length a weakly compact cardinal κ, alternating between adding reals and collapsing ω2 to have size ω1. Such an iteration P would be proper, κ-c.c., collapse κ to become ω2, and satisfy that 2 = ω2. The fact that stationary set reflection holds in any generic extension V [G] by P follows from the ability to extend an elementary embedding j with critical point κ after forcing with the proper forcing j(P)/G over V [G]. Consider the problem of obtaining a model satisfying stationary set reflection at ω2 together with 2 ω > ω2. Since in that case not all reals would be added by the iteration collapsing κ to become ω2, extending the elementary embedding becomes more difficult. Indeed, in the model referred to in the previous paragraph, a stronger stationary set reflection principle holds, namely WRP(ω2), which asserts that any stationary subset of [ω2] ω reflects to [β] for some uncountable β < ω2. By a result of Todorc̆ević, WRP(ω2) implies 2 ω ≤ ω2 (see [4, Lemma 2.9]). In this paper we demonstrate that the cardinality of the continuum provides a natural separation between ordinary stationary set reflection and higher order Date: December 2017. 2010 Mathematics Subject Classification: Primary 03E35; Secondary 03E40.