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Two Applications of Possibility Structures to the Philosophy of Mathematics| title | Two Applications of Possibility Structures to the Philosophy of Mathematics |
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| start_date | 2025/05/19 |
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| schedule | 17h-19h |
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| online | no |
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| location_info | salle de conférence de l'IHPST |
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| summary | Possibility structures provide a semantics for classical logic that generalizes Tarskian model theory. The core feature of possibility structures is that formulas are evaluated in a partially ordered sets of viewpoints, rather than at a single point. Intuitively, any such viewpoint can be thought of as a partial approximation of the full structure. In this talk, I will present two applications of possibility structures to the philosophy of mathematics. In both cases, the starting point is an impossibility result, stating that no Tarskian structure can satisfy certain conditions. As I will argue, these conditions, however, become satisfiable once we consider possibility structures.
The first application is related to non-Cantorian notions of size for infinite sets of natural numbers. One can easily show that no Tarskian elementary extension of the natural numbers can be used to assign size relationships between sets of natural numbers in a way that 1) preserves the Euclidean intuition that a set is always larger than any of its proper subsets 2) is invariant under permutations of the set of natural numbers. By contrast, I will show how to construct such an assignment of sizes using possibility structures.
The second application is related to the semantics of second-order logic. Hale has recently proposed a “deflationist” conception of properties, according to which a property is simply whatever can be referred to by a predicate is some language. This conception motivates an alternative to the standard, full semantics of second-order logic which, according to Hale, still allows Dedekind’s proof of the categoricity of arithmetic to go through. Hale’s description of this semantics, however, remains informal. I will argue that the best way to develop a formal framework that is faithful to Hale’s ideas is to work with second-order possibility structures rather than Tarskian structures. |
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| responsibles | Poggiolesi, Antonutti Marfori |
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Workflow history| from state (1) | to state | comment | date |
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| submitted | published | | 2025/05/14 07:11 UTC |
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