Yeah, sorry that was unclear; there’s no need for any form of hypercomputation to get an enumeration of the axioms of U. But you need a halting oracle to distinguish between the axioms and non-axioms. If you don’t care about distinguishing axioms from non-axioms, but you do want to get an assignment of truthvalues to the atomic formulas Q(i,j) that’s consistent with the axioms of U, then that is applying a consistent guessing oracle to U.
Yeah, sorry that was unclear; there’s no need for any form of hypercomputation to get an enumeration of the axioms of U. But you need a halting oracle to distinguish between the axioms and non-axioms. If you don’t care about distinguishing axioms from non-axioms, but you do want to get an assignment of truthvalues to the atomic formulas Q(i,j) that’s consistent with the axioms of U, then that is applying a consistent guessing oracle to U.