Regressive versions of Hindman's Theorem
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866913888390348800 |
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| author | Carlucci, Lorenzo Mainardi, Leonardo |
| author_facet | Carlucci, Lorenzo Mainardi, Leonardo |
| contents | When the Canonical Ramsey's Theorem by Erdős and Rado is applied to regressive functions one obtains the Regressive Ramsey's Theorem by Kanamori and McAloon. Taylor proved a "canonical" version of Hindman's Theorem, analogous to the Canonical Ramsey's Theorem. We introduce the restriction of Taylor's Canonical Hindman's Theorem to a subclass of the regressive functions, the $λ$-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman's Theorem and of natural restrictions of it.
In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that this same principle strongly computably reduces the well-ordering-preservation principle for base-$ω$ exponentiation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_08554 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Regressive versions of Hindman's Theorem Carlucci, Lorenzo Mainardi, Leonardo Logic Combinatorics 03D80, 05P10, 03F35 When the Canonical Ramsey's Theorem by Erdős and Rado is applied to regressive functions one obtains the Regressive Ramsey's Theorem by Kanamori and McAloon. Taylor proved a "canonical" version of Hindman's Theorem, analogous to the Canonical Ramsey's Theorem. We introduce the restriction of Taylor's Canonical Hindman's Theorem to a subclass of the regressive functions, the $λ$-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman's Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that this same principle strongly computably reduces the well-ordering-preservation principle for base-$ω$ exponentiation. |
| title | Regressive versions of Hindman's Theorem |
| topic | Logic Combinatorics 03D80, 05P10, 03F35 |
| url | https://arxiv.org/abs/2207.08554 |