The set of all analytically definable sets of natural numbers can be defined analytically

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.