38,459 research outputs found
Non-elementary proper forcing
We introduce a simplified framework for ord-transitive models and Shelah's
non elementary proper (nep) theory. We also introduce a new construction for
the countable support nep iteration
Class forcing, the forcing theorem and Boolean completions
The forcing theorem is the most fundamental result about set forcing, stating
that the forcing relation for any set forcing is definable and that the truth
lemma holds, that is everything that holds in a generic extension is forced by
a condition in the relevant generic filter. We show that both the definability
(and, in fact, even the amenability) of the forcing relation and the truth
lemma can fail for class forcing. In addition to these negative results, we
show that the forcing theorem is equivalent to the existence of a (certain kind
of) Boolean completion, and we introduce a weak combinatorial property
(approachability by projections) that implies the forcing theorem to hold.
Finally, we show that unlike for set forcing, Boolean completions need not be
unique for class forcing
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
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