Solovay will investigate various topics in Mathematical Logic. A common theme of many of the questions to be explored is the notion of the consistency strength of formal systems. Among the questions to be considered are the following: Did Godel have a proof of the independence of the axiom of choice from the axioms of type-theory prior to Cohen's discovery of forcing? Is the non-orthodox system of set-theory proposed by Quine consistent? Can one get significantly more precise estimates on the "consistency strength" of subsystems of second order number theory? One of the important discoveries of twentieth century logic is that the kinds of reasoning carried out by mathematicians can be codified into axiom systems. The most popular such system, by far, is the axiom system proposed by Zermelo (and later modified and corrected by Fraenkel and Skolem), but there are interesting extensions that have been extensively studied by set theorists as well as some significantly weaker ones beloved by the mathematically cautious. It has become clear that the study of such weak subsystems of set-theory is intimately related to large cardinals (a research specialty of Solovay's). Solovay hopes to apply his expertise in large cardinals to achieve a more precise calibration of the strength of these weak subsystems. If successful, this will bring substantial progress to a research program of some fifty years duration, which excites renewed interest in the present era of computers.
