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The set of all analytically definable sets of natural numbers can be defined analytically
Izvestiya. Mathematics , Tome 15 (1980) no. 3, pp. 469-500.

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The author proves consistency with ZFC of the following assertion: the set of all analytically definable sets $x\subseteq\omega$ is analytically definable. A subset $x$ of $\omega$ is said to be analytically definable if $x$ belongs to one of the classes $\Sigma_n^1$ of the analytic hierarchy. The same holds for $X\subseteq\mathscr P(\omega)$. Thus Tarskii's problem on definability in the theory of types is solved for the case $p=1$. The proof uses the method of forcing, with the aid of almost disjoint sets. Bibliography: 14 titles. 
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V. G. Kanovei. The set of all analytically definable sets of natural numbers can be defined analytically. Izvestiya. Mathematics , Tome 15 (1980) no. 3, pp. 469-500. http://geodesic.mathdoc.fr/item/IM2_1980_15_3_a2/

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