The Search for New Axioms in the Hyperuniverse Programme

The Hyperuniverse Programme, introduced in Arrigoni and Friedman, fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the ‘maximal iterative concept’, and the programme identifies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements in countable transitive models, the collection of which constitutes the ‘hyperuniverse’, has remarkable first-order consequences, some of which we review in Sect. 10.5.