 Gödel’s proof calculus is that of Hilbert and Ackermann’s text. An expression is in normal form if all the quantifiers occur at the beginning. The degree of an expression or formula is the number of alternating blocks of quantifiers at the beginning of the formula, assumed to begin with universal quantifiers. Gödel shows that if the completeness theorem holds for formulas of degree k it must hold for formulas of degree k + 1. Thus the question of completeness reduces to formulas of degree 1. That is, it is to be shown that any normal formula (Q)φ of degree 1 is either satisfiable or refutable, where “(Q)” stands for a (non-empty) block of universal quantifiers followed by a (possibly empty) block of existential ones.