超度的格初始段
Lattice initial segments of the hyperdegrees
AI总结：
本文证实了Sacks [1972]的猜想，通过新的格表示定理和力迫概念，证明了每个可数分配格同构于超度 $\mathcal{D}_{h}$ 的初始段，并得出了其量词理论的可判定性与不可判定性结论。
摘要：
我们通过证明每个可数分配格同构于超度 $\mathcal{D}_{h}$ 的一个初始段，证实了 Sacks [1972] 的猜想。事实上，我们证明任何超算术格的子格（特别是每个可数局部有限格）都同构于 $\mathcal{D}_{h}$ 的一个初始段。推论包括 $\mathcal{D}_{h}$ 的二量词理论的可判定性及其三量词理论的不可判定性。证明中的关键工具是一个新的格表示定理，它提供了一个力迫概念，据此我们可以在超算术背景下证明融合引理的一个版本，从而保持 $ω_{1}^{CK}$。令人惊讶的是，这种力迫的集合论类比并不保持 $ω_{1}$。另一方面，我们构造了不同构于 $\mathcal{D}_{h}$ 初始段的可数格。
英文摘要：
We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $% \mathcal{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $ω_{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $ω_{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
