On the Nonemptiness of Classes in Axiomatic Set Theory

Theorems are proved on the consistency with , for , of each of the following three propositions: (1) there exists an L-minimal (in particular, nonconstructive) such that and , but every of class with constructive code is itself constructive; (2) there exist such that their -degrees differ by a formula from , but not by formulas from with constants from ( and are said to differ by a formula ; (3) there exists an infinite, but Dedekind finite, set of class , whereas there are no such sets of class . The proof uses Cohen's forcing method.Bibliography: 17 titles.