AI 中文总结
本文解答了关于$E_0$-类泛型等价的两个问题，纠正了原条件表述，并构造了满足修正条件的Prikry型及更广泛例子，同时证明条件蕴含不可达基数。
AI 中文摘要
我们解答了文献[Tianyuan2026]中的问题8.1和8.3。首先指出，该文献中归因于普通Prikry forcing的条件按原文表述并不正确：恰当的表述应使用泛型序列值域的有限对称差，而非同一坐标上的最终相等。我们针对两种表述都给出了问题的解答。长度为$\omega$的Magidor forcing提供了一个真正的Prikry型例子，满足两个条件。刚性实数添加forcing提供了更广泛的进一步例子来源，而Jech和Shelah的一个forcing表明该条件甚至蕴含不可达基数。Kanovei和Lyubetsky的一个$E_0$-不变的Jensen型forcing给出了更强的非平凡例子，其中固定扩张中的泛型实数恰好构成一个完整的$E_0$-类。最后，一个简单的重编码将前述实数forcing例子转化为修正条件下问题版本的解答。
英文摘要
We answer Problems 8.1 and 8.3 from \cite{Tianyuan2026}. We first note that the condition attributed there to ordinary Prikry forcing is not correct as written: the appropriate formulation uses finite symmetric difference of the ranges of the generic sequences, rather than eventual equality at the same coordinates. We record answers to the two problems for both formulations. A length-$ω$ Magidor forcing gives a genuinely Prikry-type example satisfying both conditions. Rigid real-adding forcing gives a broad source of further examples, and a forcing of Jech and Shelah shows that the condition does not carry any large cardinal strength. An $E_0$-invariant Jensen-type forcing of Kanovei and Lyubetsky gives a stronger nontrivial example in which the generic reals in a fixed extension form exactly one full $E_0$-class. Finally, a simple recoding turns the real-forcing previous examples into answers for the version of the problems with corrected condition.
Commentsfixed a typo in the abstract. A "not" was left out in v1
