On sequential theorems in Reverse Mathematics
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911804447260672 |
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| author | Normann, Dag Sanders, Sam |
| author_facet | Normann, Dag Sanders, Sam |
| contents | Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence of solutions. The original and sequential theorem can often be proved via the same (or similar) proof and often have the same (or similar) logical properties, esp. if everything is formulated in the language of second-order arithmetic. In this paper, we identify basic theorems of third-order arithmetic, e.g. concerning semi-continuous functions, such that the sequential versions have very different logical properties. In particular, depending on the constructive status of the original theorem, very different and independent choice principles are needed. Despite these differences, the associated Reverse Mathematics, working in Kohlenbach's higher-order framework, is rather elegant and is still based at the core on weak König's lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13143 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On sequential theorems in Reverse Mathematics Normann, Dag Sanders, Sam Logic 03B30, 03F35 Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence of solutions. The original and sequential theorem can often be proved via the same (or similar) proof and often have the same (or similar) logical properties, esp. if everything is formulated in the language of second-order arithmetic. In this paper, we identify basic theorems of third-order arithmetic, e.g. concerning semi-continuous functions, such that the sequential versions have very different logical properties. In particular, depending on the constructive status of the original theorem, very different and independent choice principles are needed. Despite these differences, the associated Reverse Mathematics, working in Kohlenbach's higher-order framework, is rather elegant and is still based at the core on weak König's lemma. |
| title | On sequential theorems in Reverse Mathematics |
| topic | Logic 03B30, 03F35 |
| url | https://arxiv.org/abs/2403.13143 |