On Recurrence Axioms

Sakaé Fuchino; Toshimichi Usuba

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arXiv · 2402.02693

On Recurrence Axioms

Sakaé Fuchino·Toshimichi Usuba

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Abstract

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.

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BibTeX RIS

Sakaé Fuchino, Toshimichi Usuba. 2025-06-20. On Recurrence Axioms. https://arxiv.org/abs/2402.02693

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