Countabilism Workshop

Title: In Defense of Countabilism
Abstract: Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons 2000 and developed by Linnebo 2010, 2013 and Linnebo and Shapiro 2019, should also be applied to CT, in a way that vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows 2015, Scambler 2021, and Pruss 2020, aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas certain of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that the theoretical advantages of Countabilism far outweigh its potential downsides.

 

Title: Two Paths to Countabilism

Abstract: Countabilism -- the idea that, Cantor's theorem notwithstanding, all sets are countable -- has a long history in philosophy of mathematics, and there are many ways the doctrine has been supported. In this presentation I will sketch two such paths of support of recent heritage -- one with roots in forcing potentialism, and the other with roots in the theory of the real numbers -- and will try to show that each can be used to strengthen the case made by the other.

 

Title: Predicativity and Countabilism

Abstract: In his classic “Systems of predicative analysis” (Feferman 1964), Solomon Feferman highlights an important aspect of predicativism, namely the view that some totalities are inherently potential. A direct connection between predicativism and potentialism is also key to Poincaré’s later work on predicativity. Building on the modal analysis of potentialism in (Linnebo and Shapiro 2018), we present an analysis of predicativism as a form of potentialism, clarify the predicativist interpretation of Cantor’s theorem, and discuss whether predicativism is a form of countabilism. 

 

Contact: Laura.Crosilla @ ifikk.uio.no.