Totally non-proper ordinals beyond L(V λ+1)

In recent work Woodin has defined new axioms stronger than I0 (the existence of an elementary embedding j from L(Vλ+1) to itself), that involve elementary embeddings between slightly larger models. There is a natural correspondence between I0 and determinacy, but to extend this correspondence in this new framework we must insist that these elementary embeddings are proper. While at first this seemed to be a common property, in this paper will be provided a model in which all such elementary embeddings are not proper. This result fills a gap in a theorem by Woodin and justifies the definition of properness