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 $\Delta_{3}^{1}$ non-constructible set of integers." I don't understand this situation.

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    $\begingroup$ 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. $\endgroup$ Jan 31, 2022 at 14:13
  • $\begingroup$ Thanks @EmilJeřábek $\endgroup$ Feb 1, 2022 at 12:21


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