Shoenfield's theorem states that any $\Pi_{3}^{1}$ sentence that holds in $V$ holds in $L$, but I know that it is consistent with ZFC that there exists a set $A\subset \mathbb{N}$ where $A\not\in L$ and $A$ is $\Delta_{3}^{1}$, so $A$ and $\mathbb N \setminus A$ are $\Pi_{3}^{1}$. But, for any integer $n$ the statments $n\in A$ and $n \not\in A$ are $\Pi_{3}^{1}$, so $A^{L}$ and $A$ must be the same set by comprehension schema, an aparent contradiction with the consistency of "there exists a $\Delta_{3}^{1}$ nonconstructible set of integers." I don't understand this situation.
$\begingroup$
$\endgroup$
2

7$\begingroup$ The $\Pi^1_3$ and $\Sigma^1_3$ formulas that define $A$ in $V$ are only equivalent in $V$, not in $L$, hence “$A^L$” does not really make sense. $\endgroup$– Emil JeřábekJan 31, 2022 at 14:13

$\begingroup$ Thanks @EmilJeřábek $\endgroup$– Ândson joséFeb 1, 2022 at 12:21
Add a comment
