关于递归公理

AI总结：

本文研究递归公理，证明紧致泛型超巨大基数蕴含该公理，并确定其一致性强度，同时指出Laver泛型最大原理整合了众多集合论公理。

摘要：

递归公理针对偏序集类$\mathcal{P}$和参数集$A$，是在ZFC语言中的一个公理模式，它断言：如果带有来自$A$的参数的一个语句被$\mathcal{P}$中的某个偏序集力迫，那么存在一个包含这些参数且满足该语句的基模型。紧致超$C^{(\infty)}$-$\mathcal{P}$-Laver泛型超巨大连续统蕴含了针对$\mathcal{P}$和$\mathcal{H}(2^{\aleph_0})$的递归公理。该假设的一致性强度可以通过我们的主要定理来确定，这些定理断言：在紧致$\mathcal{P}$-泛型超巨大基数$\kappa$下，最小基模型（基岩）存在，并且基岩中的$\kappa$是真正的超巨大基数，甚至如果$\kappa$是紧致超$C^{(\infty)}$-$\mathcal{P}$-Laver泛型超巨大可定义基数，则$\kappa$是超$C^{(\infty)}$超巨大基数。Laver泛型最大原理（LGM）是我们语境中最强的公理组合之一，它实际上将所有已知的集合论原理和公理整合于自身，要么作为其推论，要么作为在宇宙的（许多）基模型中成立的定理。例如，Martin最大原理的双加版本是LGM的推论，而Cichoń最大原理在LGM下是宇宙的许多基模型中的一种现象。

英文摘要：

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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