A potentialist conception of ultrafinitism
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915659355521024 |
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| author | Hamkins, Joel David |
| author_facet | Hamkins, Joel David |
| contents | I shall explore various senses in which ultrafinitism can be fruitfully understood as engaging with a potentialist perspective in mathematics. First, I explain that every model $M$ of the theory of finite arithmetic -- arithmetic with a largest number, in which addition and multiplication are merely partial functions -- is bi-interpretable with a strictly taller model $M^+$, in which the arithmetic operations on objects taken from the original base model $M$ are totally defined in the extended world $M^+$. More generally, I explain how ultrafinitist ideas emerge in the modal potentialist system consisting of all models of arithmetic under end-extension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06564 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A potentialist conception of ultrafinitism Hamkins, Joel David Logic I shall explore various senses in which ultrafinitism can be fruitfully understood as engaging with a potentialist perspective in mathematics. First, I explain that every model $M$ of the theory of finite arithmetic -- arithmetic with a largest number, in which addition and multiplication are merely partial functions -- is bi-interpretable with a strictly taller model $M^+$, in which the arithmetic operations on objects taken from the original base model $M$ are totally defined in the extended world $M^+$. More generally, I explain how ultrafinitist ideas emerge in the modal potentialist system consisting of all models of arithmetic under end-extension. |
| title | A potentialist conception of ultrafinitism |
| topic | Logic |
| url | https://arxiv.org/abs/2512.06564 |