Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All Π2 consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus ∆0-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 α-Mahlo models and Π1-indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form Loc(ZFC+φ) 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.