predicate symbol (set denoted by "val") is recursively isomorphic with the corresponding set of gödel numbers of valid (full) second-order sentences in a vocabulary that contains one binary predicate symbol (set denoted by "val"). furthermore, väänänen showed that "val" is the complete π-definable set of integers, and that it is "val" not in formula_427 for any finite "m" and "n". väänänen (2007, pp. 136–139) summarizes the complexity results as follows: