When an Equivalence Relation with All Borel Classes will be Borel Somewhere
When an Equivalence Relation with All Borel Classes will be Borel Somewhere
复制标题
当与所有 Borel 类的等价关系在某处为 Borel 时
DOI：
--
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
M. Magidor
中科院分区：
文献类型：
--
作者：
William Chan;M. Magidor
In $mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E in L(mathbb{R})$ on $mathbb{R}$ with all $mathbf{Delta}_1^1$ classes and every $sigma$-ideal $I$ on $mathbb{R}$ so that the associated forcing $mathbb{P}_I$ of $I^+$ $mathbf{Delta}_1^1$ subsets is proper, there exists some $I^+$ $mathbf{Delta}_1^1$ set $C$ so that $E upharpoonright C$ is a $mathbf{Delta}_1^1$ equivalence relation. In $mathsf{ZF} + mathsf{DC} + mathsf{AD}_mathbb{R} + V = L(mathscr{P}(mathbb{R}))$, for every equivalence relation $E$ on $mathbb{R}$ with all $mathbf{Delta}_1^1$ classes and every $sigma$-ideal $I$ on $mathbb{R}$ so that the associated forcing $mathbb{P}_I$ is proper, there is some $I^+$ $mathbf{Delta}_1^1$ set $C$ so that $E upharpoonright C$ is a $mathbf{Delta}_1^1$ equivalence relation.
In $mathsf{ZFC}$, if there is a measurable cardinal with infinitely many Woodin cardinals below it, then for every equivalence relation $E in L(mathbb{R})$ on $mathbb{R}$ with all $mathbf{Delta}_1^1$ classes and every $sigma$-ideal $I$ on $mathbb{R}$ so that the associated forcing $mathbb{P}_I$ of $I^+$ $mathbf{Delta}_1^1$ subsets is proper, there exists some $I^+$ $mathbf{Delta}_1^1$ set $C$ so that $E upharpoonright C$ is a $mathbf{Delta}_1^1$ equivalence relation. In $mathsf{ZF} + mathsf{DC} + mathsf{AD}_mathbb{R} + V = L(mathscr{P}(mathbb{R}))$, for every equivalence relation $E$ on $mathbb{R}$ with all $mathbf{Delta}_1^1$ classes and every $sigma$-ideal $I$ on $mathbb{R}$ so that the associated forcing $mathbb{P}_I$ is proper, there is some $I^+$ $mathbf{Delta}_1^1$ set $C$ so that $E upharpoonright C$ is a $mathbf{Delta}_1^1$ equivalence relation.