Strong Axioms of Infinity and Elementary Embeddings

This is the expository paper on strong axioms of infinity and elementary embeddings origi;,ally to have been authored by Reinhardt and Solovay. It has been owed for some time and already cited with some frequency in the most recent set theoretical literature. However, for various reasons the paper did not appear in print far several years. The impetus for actual publication came from a series of lectures on the subject by t(anamori (Cambridge, 1975) and a set of notes circulated thereafter. Thus, although this present exposition it a detailed reworking of these notes, the basic conceptual framework was first developed by Reinhardt and 5;91ovay some years ago. One factor which turns this delay in publication to advantage is that a more comprehensive view of the concepts discussed is now possible wit1" the experience of the last few years, particularly in view of recent consistency results and also consequences in the presence of the axiom of detenninacy, A projected sequel by Solovay to this paper will deal further with these considerations. One of the most notable characteristics of the axiom of infinity is that its V~uth implies its independence of the other axioms. This, of course, is because the (infinite) set of hereditarily finite sets forms a model of the other axioms, in which there is no infinite set. Clearly, accepting an assertion whose truth implies its independence of given axioms requires the acceptance of new axioms. It is not surprising that the axiom of infinity should have this character (one would expect to have to adopt it as an axiom anyway), and moreover one would expect the existence of laIger and larger cardinalities to have such character, as indeed it has. The procedures for generating cardinals studied by Mahlo [29] provided a notable example. It is remarkable that the new consequences of the corresponding (generalized) axioms of infinity also include arithmetic statements: this application of G6del's second theorem is by now quite familiar. It is also remarkable that