Dopico, Pablo (2023) A defence of Isaacson's thesis, or how to make sense of the boundaries of finite mathematics. [Preprint]
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A defence of Isaacson's thesis, or how to make sense of the boundaries of finite mathematics.pdf Download (467kB) | Preview |
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A defence of Isaacson's thesis, or how to make sense of the boundaries of finite mathematics.pdf Download (468kB) | Preview |
Abstract
Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson's thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can be overcome. We introduce the concept of purity for arithmetical theories: an arithmetical theory is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson's thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and consistency statements—are seemingly not arithmetical in Isaacson's sense, and hence that Isaacson's thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson's thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε_0. To this end, we show that the PA-proof of such claims exclusively employs resources from finite mathematics, and that ordinals below ε_0 are finitary objects despite being infinite.
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| Item Type: | Preprint | ||||||
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| Keywords: | Isaacson's thesis, Peano Arithmetic, arithmetical truth, finite mathematics | ||||||
| Subjects: | Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Proof Specific Sciences > Mathematics |
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| Depositing User: | Mr Pablo Dopico | ||||||
| Date Deposited: | 03 Jan 2024 22:56 | ||||||
| Last Modified: | 03 Jan 2024 22:56 | ||||||
| Item ID: | 22904 | ||||||
| Subjects: | Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Proof Specific Sciences > Mathematics |
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| Date: | 30 December 2023 | ||||||
| URI: | https://philsci-archive.pitt.edu/id/eprint/22904 |
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