On Recurrence Axioms
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909654388310016 |
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| author | Fuchino, Sakaé Usuba, Toshimichi |
| author_facet | Fuchino, Sakaé Usuba, Toshimichi |
| contents | The Recurrence Axiom for a class $\mathcal{P}$ of \pos\ and a set $A$ of parameters is an axiom scheme in the language of ZFC asserting that if a statement with parameters from $A$ is forced by a poset in $\mathcal{P}$, then there is a ground containing the parameters and satisfying the statement.
The tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge continuum implies the Recurrence Axiom for $\mathcal{P}$ and $\mathcal{H}(2^{\aleph_0})$. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly $\mathcal{P}$-generic hyperhuge cardinal $κ$, and that $κ$ in the bedrock is genuinely hyperhuge, or even super $C^{(\infty)}$ hyperhuge if $κ$ is a tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge definable cardinal.
The Laver Generic Maximum (LGM), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For example, double plus version of Martin's Maximum is a consequence of LGM while Cichoń's Maximum is a phenomenon in many grounds of the universe under LGM. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02693 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Recurrence Axioms Fuchino, Sakaé Usuba, Toshimichi Logic 03E45, 03E50, 03E55, 03E57, 03E65 The Recurrence Axiom for a class $\mathcal{P}$ of \pos\ and a set $A$ of parameters is an axiom scheme in the language of ZFC asserting that if a statement with parameters from $A$ is forced by a poset in $\mathcal{P}$, then there is a ground containing the parameters and satisfying the statement. The tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge continuum implies the Recurrence Axiom for $\mathcal{P}$ and $\mathcal{H}(2^{\aleph_0})$. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly $\mathcal{P}$-generic hyperhuge cardinal $κ$, and that $κ$ in the bedrock is genuinely hyperhuge, or even super $C^{(\infty)}$ hyperhuge if $κ$ is a tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge definable cardinal. The Laver Generic Maximum (LGM), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For example, double plus version of Martin's Maximum is a consequence of LGM while Cichoń's Maximum is a phenomenon in many grounds of the universe under LGM. |
| title | On Recurrence Axioms |
| topic | Logic 03E45, 03E50, 03E55, 03E57, 03E65 |
| url | https://arxiv.org/abs/2402.02693 |