Forcing in Contemporary Philosophy of Set Theory

The project Forcing in Contemporary Philosophy of Set Theory explores the philosophical implications of the mathematical technique of forcing. Forcing was developed in the 1960s in the mathematical field of set theory to deal with the consequences of Kurt Gödels famous work on undecidable statements. It turned out to be a most powerful and influential tool whose results changed set theory dramatically; forcing is one of the most important techniques in set theory up to this date. As set theory plays an important role in questions relating to the foundations of mathematics, the philosophical implications of forcing are of great interest. So far, the focus of current research programs in the philosophy of set theory has been on the philosophical influence of the results forcing provides us with. The conclusions that these programs draw vary widely and the discussion between them dominates the current debate in the philosophy of set theory. This project wants to provide a contribution to this debate by observing that the differing conclusions of the programs are not just determined by the results of forcing, but also by the way in which these results are obtained, namely by using the forcing technique in various ways. Building on this observation, the main research hypothesis of the project is that the method of forcing itself and the way(s) in which it is used by set theorists is one of the differentiating factors responsible for the philosophical conclusions that they draw in these programs. To analyze this situation the project has to develop new set-theoretic procedures which in turn can be used to solve philosophical questions. This methodological approach is part of mathematical philosophy, a philosophical field in which philosophical questions are explicated by using tools from mathematics. Working in this interdisciplinary field of mathematics and philosophy, the overall goal of the project is to deliver the first general study of forcing in contemporary philosophy of set theory.