Abstract

Abstract. We produce counterexamples to the unique and cofinal branches hypotheses, assuming (slightly less than) the existence of a cardinal which is strong past a Woodin cardinal. Martin–Steel [3] introduced the notion of an iteration tree, and with it the question of iterability: the existence of a method for choosing cofinal branches through iteration trees in a manner that secures the wellfoundedness of the models generated by the trees. The question has since become central to the study of large cardinals, with particular importance to the construction of inner models and to proofs of determinacy. (For specific examples see Neeman [4] and Neeman [5].) Martin–Steel [3] suggested a natural hypothesis, which if true, would combine with the results they proved to yield iterability for countable elementary substructures of rank initial segments of V, enough for the applications to proofs of determinacy, and some simple applications to inner model theory. (In the deeper parts of inner model theory, one wants to construct iterable models satisfying large cardinal hypotheses in situations in which there may be no large cardinals in V. Here the possibly vacuous iterability of elementary submodels of V is irrelevant.) The hypothesis, known as the Unique Branches Hypothesis, or UBH for short, states that every countable iteration tree on V has at most one cofinal wellfounded branch. Combined with the results of Martin–Steel [3] it would imply that in fact every countable iteration tree on V which is sufficiently closed has exactly one cofinal wellfounded branch. Martin–Steel [3] also coined the Cofinal Branches Hypothesis, CBH for short, stating that every countable iteration tree on V has at least one cofinal wellfounded branch. For sufficiently closed iteration trees this is a consequence of UBH. Woodin refuted UBH about two years after it had been introduced, using very large cardinals involving embeddings of a rank initial segment of the universe into itself. In 1999, Woodin went on to refute CBH under the weaker hypothesis that there is a supercompact cardinal with a Woodin cardinal above it; this gave a counterexample to UBH under the same hypothesis. The counterexample to UBH, which Woodin presented at the meeting on Core Model Theory at