一致化与内部绝对性
Uniformization and Internal Absoluteness
摘要：
关于理想的可测性与某些力迫概念的绝对性原理紧密相连。我们研究了一个均匀化原理，假设存在一个均匀化函数的一个大的集合，相对于一个给定的理想。证明了对于所有$σ$ -理想$I$，使得模$I$的Borel集的理想强制$\mathbb{P}_I$是适当的，这个一致化原理等价于关于$\mathbb{P}_I$的投影公式的绝对性原理，我们称之为内部绝对性.此外，我们证明了偏序集$\mathbb{P}_I$关于$I$的可测性与$1$ -步绝对性等价.这些等价是新的，甚至科恩和随机强迫，他们是，据我们所知，第一个精确的正则性和绝对性之间的等价超出了第二层次的射影层次。
英文摘要：
Measurability with respect to ideals is tightly connected with absoluteness principles for certain forcing notions. We study a uniformization principle that postulates the existence of a uniformizing function on a large set, relative to a given ideal. We prove that for all $σ$-ideals $I$ such that the ideal forcing $\mathbb{P}_I$ of Borel sets modulo $I$ is proper, this uniformization principle is equivalent to an absoluteness principle for projective formulas with respect to $\mathbb{P}_I$ that we call internal absoluteness. In addition, we show that it is equivalent to measurability with respect to $I$ together with $1$-step absoluteness for the poset $\mathbb{P}_I$. These equivalences are new even for Cohen and random forcing and they are, to the best of our knowledge, the first precise equivalences between regularity and absoluteness beyond the second level of the projective hierarchy.
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