| Résume | The discovery of forcing in the 60's revolutionized set theory by providing a flexible and controlled way to extend models of Zermelo-Fraenkel (ZF). While not every extension is literally obtained by forcing, there is a strong sense in which it pivotal in understanding the structure of models as a whole. One remarkable result for instance is that, in the sense of forcing, every model of ZFC is close to its inner model of hereditarily ordinal definable sets (HOD). Another one is the Intermediate Model Theorem which states that any model of ZFC, intermediate to a "ground model" and one of its forcing extensions, is itself a forcing extension of such ground. This completely fails if we only assume ZF instead (i.e. set theory without the Axiom of Choice). Can something still be said? It turns out that the answer is related to a particular class of choice principles, of which its easiest form was introduced by Kinna and Wagner in the 50's, which gives a full picture of the situation. The goal of our talk is to present these results, by giving all necessary details to understand what they mean to a general logic audience. This is joint work with A. Karagila. |
