UC Berkeley

In support of the Inner Model Program, we establish that a strengthening of well-foundedness, strong well-foundedness, holds for models constructed with iteration trees of certain important types. We subsequently find instances in which this property implies that every countable subset of a model is generic over the model, thereby proving instances of the Genericity Hypothesis. We also place the property of strong well-foundedness in context by proposing how it could be used to establish iterability, a key concept of the Inner Model Program, via a new route.