181 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
Localizing the axioms
We examine what happens if we replace ZFC with a localistic/relativistic
system, LZFC, whose central new axiom, denoted by , says that
every set belongs to a transitive model of ZFC. LZFC consists of plus some elementary axioms forming Basic Set Theory (BST). Some
theoretical reasons for this shift of view are given. All consequences
of ZFC are provable in . LZFC strongly extends Kripke-Platek (KP)
set theory minus -Collection and minus -induction scheme.
ZFC+``there is an inaccessible cardinal'' proves the consistency of LZFC. In
LZFC we focus on models rather than cardinals, a transitive model being
considered as the analogue of an inaccessible cardinal. Pushing this analogy
further we define -Mahlo models and -indescribable models, the
latter being the analogues of weakly compact cardinals. Also localization
axioms of the form are considered and their global
consequences are examined. Finally we introduce the concept of standard compact
cardinal (in ZFC) and some standard compactness results are proved.Comment: 38 page
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