6 research outputs found
Disjoint Stationary Sequences on an Interval of Cardinals
We introduce strong distributivity, a strengthening of distributivity, which
implies preservation of ccness and stationarity, afterwards showing a stronger
version of the Easton Lemma. We also introduce a new framework for working with
arbitrary orders on products of sets. Both concepts are applied together to
answer two questions of Krueger using a new version of Mitchell's Forcing.Comment: 25 pages, 0 figure
Slender Trees and the Approximation Property
We obtain a relatively simple criterion for when a forcing has the
-approximation property, generalizing a result of Unger.
Afterwards we apply this criterion to construct variants of Mitchell Forcing in
order to answer questions posed by Mohammadpour.Comment: 28 pages, 0 figure
Set-Theoretic Geology
A ground of the universe V is a transitive proper class W subset V, such that
W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G]
for some W-generic filter G subset P in W . The model V satisfies the ground
axiom GA if there are no such W properly contained in V . The model W is a
bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle
of V is the intersection of all grounds of V . The generic mantle of V is the
intersection of all grounds of all set-forcing extensions of V . The generic
HOD, written gHOD, is the intersection of all HODs of all set-forcing
extensions. The generic HOD is always a model of ZFC, and the generic mantle is
always a model of ZF. Every model of ZFC is the mantle and generic mantle of
another model of ZFC. We prove this theorem while also controlling the HOD of
the final model, as well as the generic HOD. Iteratively taking the mantle
penetrates down through the inner mantles to what we call the outer core, what
remains when all outer layers of forcing have been stripped away. Many
fundamental questions remain open.Comment: 44 pages; commentary concerning this article can be made at
http://jdh.hamkins.org/set-theoreticgeology