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Inverse limit reflection and the structure of L(Vλ+1)

We extend the results of Laver on using inverse limits to reflect large cardinals of the form, there exists an elementary embedding Lα(Vλ+1) → Lα(Vλ+1). Using these inverse limit reflection embeddings directly and by broadening the collection of U(j)-representable sets, we prove structural results of L(Vλ+1) under the assumption that there exists an elementary embedding j : L(Vλ+1) → L(Vλ+1). As a consequence we show the impossibility of a generalized inverse limit X-reflection result for X ⊆ Vλ+1, thus focusing the study of L(ℝ) generalizations on L(Vλ+1).