In "Class and Membership" (2003) Kit Fine proposed a Copernican Revolution in the theory of classes. In the iterative conception of set one takes the set-membership relation for granted and one iteratively expands the ontology of sets. In the Copernican approach to classes one takes the ontology of classes for granted and one iteratively expands the class-membership relation. A striking feature of this theory is that it allows a universal (and thus self-membered) class while retaining classical logic. Related theories have been explored by Button, Church, Forster, Linnebo, and Roberts.
This workshop brings together leading researchers to assess the state of the Copernican Revolution.
Provisional Schedule
Friday May 30:
Saturday May 31.
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Abstracts:
Kit Fine, “Classes and A-Objects”
I explain the intimate connection between Copernican class theory and the theory of A-objects and how the construction of Copernican classes might be extended to A-objects.
Øystein Linnebo, “Constructing untyped properties”
Orthodox “Copernicanism” faces various challenges. To overcome these, I explore a constructional approach to untyped properties, starting with the Vicious Circle Principle and gradually liberalizing. The exploration identifies three constructional routes to (a version of) the allegedly “Copernican” theory.
Sam Roberts, “How to be a Copernican about Properties”
The Copernican approach was first proposed by Kit Fine—who applied the framework to classes—and independently by Øystein Linnebo—who applied it to properties. In this talk, I’ll explore its application to properties. I’ll begin by extracting a minimal Copernican theory and then raise what I take to be the most pressing unresolved questions. I’ll examine several answers to these questions that have been put forward and argue that they lead to importantly different views.
Gabriel Uzquiano, “Iterative Divisibility”
The Copernican approach to classes takes the ontology of classes for granted and embarks in a journey of ideological expansion, whereby one iteratively generates successive reinterpretations of the class-theoretic vocabulary which acknowledge as classes some of what prior interpretations had treated as urelements.
We will explore the prospects of a Copernican approach to the part to whole relation. We take the universe of all objects for granted and ask how to decompose it into parts. Given a minimal expansion of the mereological framework, we deploy a variant of Russell's paradox to suggest that there is no maximal decomposition of the universe into parts. No matter what a putative interpretation of the mereological vocabulary may be, we may iteratively generate successive reinterpretations of the vocabulary which come to acknowledge as complex parts some of what prior interpretations had treated as mereological atoms. The iteration supports an open-ended series of ever more fine-grained decompositions of the universe into parts in which successive members of the series come to draw mereological distinctions to which prior members had been blind.
One application of the framework is the articulation of a mereological view of classes on which they are conceived as parts of the set-theoretic universe. There is, however, no maximal decomposition of the domain of set theory into classes but rather an open-ended series of decompositions that support a variety of ever more powerful class comprehension principles in line with the original Copernican approach to classes.
Jon Litland, “Copernican Ground”
Many grounding theorists tacitly assume structured propositions holding that the proposition Pa is identical to the proposition Qb iff P=Q and a=b. But this is couter-cantorian: it allows us to construct an injection from the properties of propositions into the propositions. Grounding theorists are on the hook for counter-cantorianism anway: standard principles about immediate ground allow us to construct an injection from properties of propositions to propositions. This paper proposes that we should accept the counter-cantorian consequences and it develops a systematic counter-cantorian theory. The idea is that properties are defined by specifying their grounding profiles, the contribution they make to how propositions formed by applying them are grounded. Properties are generated in a hierarchy of orders. To avoid paradox we take the “Copernican Turn” and impose a restriction on the use of the grounding relation in defining grounding profiles: in defining grounding profiles at some order o we are restricted to using the grounding-relation as it has been defined up through order o. We then use the theory to throw new light on some of the following issues: absolutely general quantification; the puzzles of ground; the Russell-Myhill Paradox; the Kaplan-Prior paradox.
Tim Button, “Levels make a Fine Church”
I will discuss three theories of sets, which provide us with a Boolean algebra of sets whilst containing ZFC. The first theory is Fine's (2005); the second is Church's (1974); the third is a slight refinement of Church's theory (which I presented in 2024). I will explain how these refinements can help further the cause of Copernican revolutionaries.