Recognizable sets and Woodin cardinals: computation beyond the constructible universe

We call a subset of an ordinal recognizable if it is the unique subset x of for which some Turing machine with ordinal time and tape and an ordinal parameter, that halts for all subsets of as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given ⌃1 formula in L[x]. We further define the recognizable closure for subsets of by closing under relative recognizability for subsets of . We prove several results about recognizable sets and their variants for other types of machines. Notably, we show the following results from large cardinals. • Recognizable sets of ordinals appear in the hierarchy of inner models at least up to the level Woodin cardinals, while computable sets are elements of L. • A subset of a countable ordinal is in the recognizable closure for subsets of if and only if it is an element of the inner model M1, which is obtained by iterating the least measure of the least fine structural inner model M1 with a Woodin cardinal through the ordinals.