2,128 research outputs found
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
The bounded proper forcing axiom and well orderings of the reals
We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(Ļ_1) which is Ī_1 definable with parameter a subset of Ļ_1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes N_2 and also satisfies BPFA must contain all subsets of Ļ_1. We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the HƤrtig quantifier is not lightface projective
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