View source for Inner model

An '''inner model''' is a transitive class containing all ordinals that satisfies the axioms of ZFC but is smaller than the full universe V. The study of inner models is the primary technique for proving relative consistency results in set theory: if a small inner model can be shown to satisfy an axiom, then the consistency of that axiom follows from the consistency of ZFC. The paradigmatic inner model is Gödel's [[Constructible Universe|constructible universe]] L, the minimal inner model containing all ordinals.

Modern inner model theory goes far beyond L. It constructs inner models that contain [[Large cardinal|large cardinals]], using sophisticated techniques of [[extender model|extender models]] and fine structure to build canonical models that satisfy increasingly strong axioms. The goal of the inner model program is to show that every natural set-theoretic hypothesis is true in some canonical inner model — a program that, if successful, would provide a kind of ultimate calibration for the consistency strength of set theory.

''The inner model program is not a retreat into smaller universes. It is a strategy of controlled exploration: by building canonical models that satisfy strong axioms, we learn what those axioms commit us to, and whether the full universe V can contain them. The constructible universe L is not a toy model; it is a probe, and the inner model program is the art of probing the unknown with the known.''

[[Category:Mathematics]]
[[Category:Logic]]