AbstractModal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, the right modal logic is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities.
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References found in this work
Science Without Numbers: A Defence of Nominalism.Hartry H. Field - 1980 - Princeton, NJ, USA: Princeton University Press.
Past, present and future.Arthur Prior - 1967 - Revue Philosophique de la France Et de l'Etranger 157:476-476.
Citations of this work
Mirroring Theorems in Free Logic.Ethan Brauer - 2020 - Notre Dame Journal of Formal Logic 61 (4):561-572.
Divergent Potentialism: A Modal Analysis With an Application to Choice Sequences.Ethan Brauer, Øystein Linnebo & Stewart Shapiro - forthcoming - Philosophia Mathematica.
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