We study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of $\Pi ^1_n$ -indescribability where $n<\omega $ to that of $\Pi ^1_\xi $ -indescribability where $\xi \geq \omega $ . By iterating Feng’s Ramsey operator [12] on the various $\Pi ^1_\xi $ -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng’s original Ramsey hierarchy. We isolate Ramsey properties which provide strictly increasing hierarchies between Feng’s $\Pi _\alpha $ -Ramsey and $\Pi _{\alpha +1}$ -Ramsey cardinals for all odd $\alpha <\omega $ and for all $\omega \leq \alpha <\kappa $ . We also show that, given any ordinals $\beta _0,\beta _1<\kappa $ the increasing chains of ideals obtained by iterating the Ramsey operator on the $\Pi ^1_{\beta _0}$ -indescribability ideal and the $\Pi ^1_{\beta _1}$ -indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of $\Pi ^1_\xi $ -indescribability and Ramseyness.

中文翻译：

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通过不可描述性对拉姆齐等级制度的改进

我们研究了与 Ramseyness 相关的大基数性质，其中要求齐次集来满足各种不可描述的超限程度。Sharpe 和 Welch [25] 以及 Bagaria [1] 独立地扩展了$\Pi ^1_n$- 难以形容的地方$n<\欧米茄$到那个$\Pi ^1_\xi $- 难以形容的地方$\xi \geq \omega $. 通过迭代 Feng 的 Ramsey 算子 [12] 在各种$\Pi ^1_\xi $-不可描述的理想，我们获得了新的大基数层次和相应的正常理想的非线性递增层次。我们提供了由此产生的理想之间的包含关系的完整说明，并表明相应的大基数属性产生了冯的原始拉姆齐层次结构的严格线性细化。我们隔离了 Ramsey 属性，这些属性提供了 Feng 的严格递增的层次结构$\Pi_\alpha $-拉姆齐和$\Pi _{\alpha +1}$- 所有奇怪的拉姆齐红雀$\alpha <\omega $并为所有人$\omega \leq \alpha <\kappa $. 我们还表明，给定任何序数$\beta _0,\beta _1<\kappa $通过在$\Pi ^1_{\beta _0}$-不可描述的理想和$\Pi ^1_{\beta _1}$-不可描述的理想分别，最终相等；此外，我们确定了这种相等发生的最小程度的 Ramseyness。作为对我们结果的应用，我们表明可以根据通用基本嵌入来描述我们新的大基本概念和相应的理想；作为一种特殊情况，这会产生以下的通用嵌入特征$\Pi ^1_\xi $-不可描述性和拉姆齐性。