Undecidable indirectly reflexive sentences

A generalization of the well-known diagonalization (reflexion) lemma for the case of indirect reflexion is proved for the theory of formal arithmetic. Indirectly reflexive sentences about provability and refutability in the $\omega$-consistent theory of formal arithmetic are studied. Existence of undecidable sentences among some sets of indirectly reflective sentences is proved. If provability and refutability are replaced by truth and falsehood, existence of undecidable sentences leads to paradoxes.