ForcingFest Vol. I: Posets

About
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 settheoretic 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 philosophydriven exposition of the topic aims at highlighting the philosophical interest of the issues that forcing aims at solving.

Info
• The course will be mainly in-person at the University of Oslo (Blindern Campus), but we are planning also to stream it via Zoom.
• Everyone is welcome to attend, but to do so, please register by contacting
davide.sutto@ifikk.uio.no or poset.group@gmail.com
• The workshop also counts as a PhD Course worth 2 ECTS, under completion of further assignments. Contact Davide Sutto for more info.

Organizers:
PoSeT Group
• Maria Beatrice Buonaguidi (King’s College London)
• Pablo Dopico (King’s College London)
• Davide Sutto
With the Kind Support of
• Øystein Linnebo and the C-FORS Project;
• The European Society for the Philosophy of Mathematics

Program:
Friday 21st of June 2024 (GMH 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 intentional models that philosophers are already interested in:
namely, possibility models.

• 13.00-14.00: Lunch

• 14.00-15.30:
Deborah Kant (University of Hamburg): Set-Theoretic Bicontextualism
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 settheoretic 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

• 16.00-17.30:
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

19.30: Dinner