Iterating symmetric extensions

The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of  between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of  . We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.