Recognizable Sets and Woodin Cardinals: Computation beyond the Constructible Universe

We call a subset of an ordinal λ recognizable if it is the unique subset x of λ for which some Turing machine with ordinal time and tape, which halts for all subsets of λ as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given Σ1 formula in L[x]. We prove several results about sets of ordinals recognizable from ordinal parameters by ordinal time Turing machines. Notably we show the following results from large cardinals. • Computable sets are elements of L, while recognizable objects appear up to the level of Woodin cardinals. • A subset of a countable ordinal λ is in the recognizable closure for subsets of λ if and only if it is an element of M∞, where M∞ denotes the inner model obtained by iterating the least measure of M1 through the ordinals, and where the recognizable closure for subsets of λ is defined by closing under relative recognizability for subsets of λ.