A defense of Isaacson's thesis, or how to make sense of the boundaries of finite mathematics
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| Format: | Recurso digital |
| Langue: | anglais |
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2024
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| _version_ | 1866902090994941952 |
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| author | Dopico, Pablo |
| author_facet | Dopico, Pablo |
| contents | <p>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 (<strong>PA</strong>) is <em>the</em> 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 theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some <strong>PA</strong>-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of <strong>PA</strong>. 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 <strong>PA</strong>-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below <span><span><span>ε0</span></span></span>. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below <span><span><span>ε0</span></span></span> are finitary objects despite being infinite.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_1007_s11229-024-04488-0 |
| institution | Zenodo |
| language | eng |
| publishDate | 2024 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A defense of Isaacson's thesis, or how to make sense of the boundaries of finite mathematics Dopico, Pablo Isaacson's thesis · Peano Arithmetic Arithmetical truth · Finite mathematics <p>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 (<strong>PA</strong>) is <em>the</em> 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 theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some <strong>PA</strong>-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of <strong>PA</strong>. 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 <strong>PA</strong>-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below <span><span><span>ε0</span></span></span>. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below <span><span><span>ε0</span></span></span> are finitary objects despite being infinite.</p> |
| title | A defense of Isaacson's thesis, or how to make sense of the boundaries of finite mathematics |
| topic | Isaacson's thesis · Peano Arithmetic Arithmetical truth · Finite mathematics |
| url | https://doi.org/10.1007/s11229-024-04488-0 |