We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every Π 1 1 set, for example, is decidable by such machines, and the semi-decidable sets form a portion of the ∆ 1 2 sets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators.

Abstract. We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 κ = κ + , another for which 2 κ = κ ++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that H V κ + ⊆ HOD W . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

Abstract. If an extension V ⊆ V satisfies the δ approximation and cover properties for classes and V is a class in V , then every suitably closed embedding j : V → N in V with critical point above δ restricts to an embedding j V amenable to the ground model V . In such extensions, therefore, there are no new large cardinals above δ. This result extends work in [Ham01].