In this 3 talk series (part 1: this part, part 2, part 3) I will give an introduction to the technique of symmetric systems and present some of the recent results we have obtained joint with A. Karagila. Symmetric systems produce so called symmetric extensions which are intermediate models between \(V\) and forcing extension \(V[G]\). These models may not satisfy the Axiom of Choice and their primary use is to obtain consistency results for \(\mathsf{ZF}\). On the other hand, they are also useful in generally understanding and classifying intermediate models.
