On the nonemptiness of classes in axiomatic set theory

Theorems are proved on the consistency with $ZF$, for $n\geqslant2$, of each of the following three propositions: (1) there exists an $L$-minimal (in particular, nonconstructive) $a\subseteq\omega$ such that $V=L[a]$ and $\{a\}\in\Pi_n^1$, but every $b\subseteq\omega$ of class $\Sigma_n^1$ with constructive code is itself constructive; (2) there exist $a,b\subseteq\omega$ such that their $L$-degrees differ by a formula from $\Pi_n^1$, but not by formulas from $\Sigma_n^1$ with constants from $L$ ($X$ and $Y$ are said to differ by a formula $\sim[(\exists\,x\in X)\varphi(x)\equiv(\exists\,y\in Y)\varphi(y)])$; (3) there exists an infinite, but Dedekind finite, set $X\in\mathscr P(\omega)$ of class $\Pi_n^1$, whereas there are no such sets of class $\underline\Sigma_n^1$. The proof uses Cohen's forcing method.
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