Recognizable sets and Woodin cardinals : Computation beyond the constructible universe

Abstract

We call a subset of an ordinal λrecognizableif it is the unique subset xof λfor which some Turing machine with ordinal time and tape and an ordinal parameter, that halts for all subsets of λas input, halts with the final state 0. Equivalently, such a set is the unique subset xwhich satisfies a given Σ₁formula in L[x]. We further define the recognizable closurefor subsets of λ by closing under relative recognizability for subsets of λ.<br /><br /><br />We prove several results about recognizable sets and their variants for other types of machines. Notably, we show the following results from large cardinals.<br /><br />•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.<br /><br />•A subset of a countable ordinal λis in the recognizable closure for subsets of countable ordinals if and only if it is an element of the inner model M^∞, which is obtained by iterating the least measure of the least fine structural inner model M₁with a Woodin cardinal through the ordinals.