Embeddings of Iteration Trees

Abstract

This paper, dating from May 1991, contains preliminary (and unpublishable) notes on investigations about iteration trees. They will be of interest only to the specialist. In the first two sections I define notions of support and embeddings for tree iterations, proving for example that every tree iteration is a direct limit of finite tree iterations. This is a generalization to models with extenders of basic ideas of iterated ultrapowers using only ultrapowers. In the final section (which is most of the paper) I sketch a proof that any tree iteration can be embedded into a normal iteration, that is, a tree iteration with the extenders in nondecreasing order of strength and with strictly increasing critical points.