ForcingFest Vol. I: Posets

Abstract: I will explain how the forcing construction can be seen as a direct implementation of the iterative conception, giving rise to the cumulative hierarchy, but undertaken in the context of multivalued logic. The shape of the logic that is available in effect enables a certain constructive interference of the truth values in such a way that can affect the truth judgements. The core utility of forcing arises from the fact that we can often control these consequences by making a careful choice of the logic to be used, thereby controlling the truth values even of natural set-theoretic statements such as the continuum hypothesis.

Abstract: As central as the method of forcing is within set theory, it has yet to be incorporated into the philosopher's toolbox. That strikes me as a shame, since it may well have important applications to a number of philosophical problems. One barrier is that typical presentations of forcing are overly dry and technical and make it seem inherently bound up with its applications within set theory. The purpose of this talk is try to rectify this by showing how forcing models can be seen as a special case of a more general class of intensional models that philosophers are already interested in: namely, possibility models.

Abstract: In set-theoretic ontology, philosophers grapple with two primary positions: universism and multiversism, which correspond to the epistemological perspectives of absolutism and pluralism. Notably, forcing plays a pivotal role in set-theoretic practice. Hamkins contends that a multiverse view is inherently tied to set-theoretic practice, while Maddy advocates for a universe view based on her analyses of the same practice. To assess these claims, I conducted an extensive interview study with set-theoretic practitioners. The findings suggest that many practitioners hold determinate views on absolutism or pluralism and recognize forcing as a fundamental aspect of their work. I will elucidate how practitioners' perspectives on forcing relate to their research areas and philosophical stances. I conclude that overarching claims about set-theoretic practice, whether rooted in universism or multiversism, presuppose a homogeneity within the set-theoretic community that doesn't align with reality. To present philosophically sound claims reflective of set-theoretic practice, one must acknowledge its inherent heterogeneity.

Abstract: Simultaneity Relativism is the familiar idea that there is no physically absolute notion of sameness-of-time, only various coordinate-relative ones. Analogously, Countability Relativism is the idea that there is no mathematically absolute notion of uncountability, only various model-relative ones. In my talk I will explore the case for countability relativism, beginning with Skolem and ending with contemporary arguments involving forcing.