2013/05/09 by Viale, Matteo
#03E40 #03E57 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1305.2058
We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces MM++ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of MM++ collapsing a superhuge cardinal to become the second uncountable cardinal produces a model in which the cutoff of the category of stationary set preserving forcings at any rank initial segment of the universe of large enough height is forcing equivalent to a presaturated tower of normal filters. We let MM+++ denote this statement and we prove that the theory of L(Ordω1) with parameters in P(ω1) is generically invariant for stationary set preserving forcings that preserve MM+++. Finally we argue that the work of Larson and Asperó shows that this is a next to optimal generalization to the Chang model L(Ordω1) of Woodin's generic absoluteness results for the Chang model L(Ordω). It remains open whether MM+++ and MM++ are equivalent axioms modulo large cardinals and whether MM++ suffices to prove the same generic absoluteness results for the Chang model L(Ordω1).