ForcingFest Vol. I: Posets

21 June 2024, University of Oslo, Oslo, Norway

Forcing is, arguably, one of the greatest milestones achieved within set theory. Despite its being around for roughly 60 years, the technique introduced by Paul Cohen has not been deeply studied by philosophers of mathematics. The topic sparked some serious interest only in the last few of years. The latest developments of a modal analysis of forcing and of set-theoretic multiversism due to Joel David Hamkins and Benedikt Löwe generated a new philosophical interest for forcing, which brought out very deep philosophical questions on the nature of the set-theoretical universe. Moreover, the surge of set-theoretic potentialism in the philosophy of set theory made forcing an essential tool for a potentialist account of the infinite.

In light of the new philosophical interest sparked around the topic in the last years, the first ForcingFest will deal with forcing (with posets) and its philosophical applications. The goal is to foster interaction between junior researchers and leading scholars in this new area of research that we may call "Philosophy of Forcing". The expected outcome of the course is twofold: on the one hand, participants are expected to get acknowledged with the basics of forcing in a way that is more suited for philosophers, with links to areas (such as modal logic) that are more familiar to them; on the other hand, the philosophy-driven exposition of the topic aims at highlighting the philosophical interest of the issues that forcing aims at solving.

SPEAKERS

The workshop consists of 4 lectures and a final general discussion.

Joel David Hamkins

University of Notre Dame

Deborah Kant

University of Hamburg

Sam Roberts

University of Konstanz

Chris Scambler

University of Oxford

PROGRAM

Venue: University of Oslo, Blindern Campus, Georg Morgenstiernes hus, Room 652. 

09.00 - 09.30: Coffee and Welcome

09.30 - 11.00: Joel David Hamkins (O'Hara Professor of Logic, University of Notre Dame), Forcing is simply the iterative conception undertaken with multivalued logic

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.

11.00 - 11.30: Break

11.30 - 13.00: Sam Roberts (University of Konstanz), Forcing for Philosophers

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.

13.00 - 14.00: Lunch

14.00 - 15.30: Deborah Kant (University of Hamburg), Forcing and models in set-theoretic practice: What do we learn about ontology?

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.

15.30 - 16.00: Break

15.30 - 16.00: Chris Scambler (University of Oxford): Forcing and Countability Relativism

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.

17.30 - 18.00: Discussion

20:30: Dinner at Dinner Restaurant 

INFO



Further information on online participation and schedule will come. 


ORGANIZERS

Davide Sutto (University of Oslo), Maria Beatrice Buonaguidi (King's College London), Pablo Dopico (King's College London).

With the kind support of ERC Advanced Project C-FORS project led by Øystein Linnebo

and with the sponsorship of the European Society for the Philosophy of Mathematics.