Reflection and Recurrence
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916456239726592 |
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| author | Fuchino, Sakaé |
| author_facet | Fuchino, Sakaé |
| contents | We examine the Zermelo Fraenkel set theory with Choice (ZFC) enhanced by one of the (structural) reflection principles down to a small cardinal and/or Recurrence Axioms defined below. The strongest forms of reflection principles spotlight the three scenarios in which the size of the continuum is either $\aleph_1$, or $\aleph_2$, or very large, while the maximal setting of Recurrence Axioms points to the set-theoretic universe with the continuum of size $\aleph_2$.
We discuss that both the Reflection Principles and Recurrence Axioms can be construed as preferable candidates of the extension of ZFC in terms of the criteria of Gödel's Program. From this view point, the maximal possible (consistent) combination of these principles and axioms, or even some natural strengthening of the combination (which we want to call ``Laver-generic Maximum'' (LGM)) may be considered as the ultimate extension of ZFC (of course ``ultimate'' only for now -- because of the Incompleteness Theorems): LGM resolves the size of the continuum to be $\aleph_2$ and integrates practically all known statements consistent with ZFC in itself either as its consequences (as it is the case with Martin's Maximum$^{++}$) or as theorems holding in many grounds of the universe (as it is the case with Cichoń's Maximum). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_20343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reflection and Recurrence Fuchino, Sakaé Logic 03E45, 03E50, 03E55, 03E57, 03E65 We examine the Zermelo Fraenkel set theory with Choice (ZFC) enhanced by one of the (structural) reflection principles down to a small cardinal and/or Recurrence Axioms defined below. The strongest forms of reflection principles spotlight the three scenarios in which the size of the continuum is either $\aleph_1$, or $\aleph_2$, or very large, while the maximal setting of Recurrence Axioms points to the set-theoretic universe with the continuum of size $\aleph_2$. We discuss that both the Reflection Principles and Recurrence Axioms can be construed as preferable candidates of the extension of ZFC in terms of the criteria of Gödel's Program. From this view point, the maximal possible (consistent) combination of these principles and axioms, or even some natural strengthening of the combination (which we want to call ``Laver-generic Maximum'' (LGM)) may be considered as the ultimate extension of ZFC (of course ``ultimate'' only for now -- because of the Incompleteness Theorems): LGM resolves the size of the continuum to be $\aleph_2$ and integrates practically all known statements consistent with ZFC in itself either as its consequences (as it is the case with Martin's Maximum$^{++}$) or as theorems holding in many grounds of the universe (as it is the case with Cichoń's Maximum). |
| title | Reflection and Recurrence |
| topic | Logic 03E45, 03E50, 03E55, 03E57, 03E65 |
| url | https://arxiv.org/abs/2410.20343 |