 To early logicians it seemed natural to use second-order logic on a par with first order logic, as if there were not much difference. This included Russell, Löwenheim, Hilbert, and Zermelo. Essentially only Skolem emphasized the difference (see Moore 1988 for the emergence of first order logic). In 1929 Gödel proved his Completeness Theorem and the next year his Incompleteness Theorem. It became gradually obvious that second-order logic is very different from first order logic. One of the first indicators was the incompleteness phenomenon. Gödel showed that any effective axiomatization of number theory is incomplete. On the other hand, there was a simple finite categorical—hence complete  (§10)—axiomatization  of the structure \((\oN,\) \(+,\) \(\times)\) in second-order logic (see also the discussion related to (\ref{ind})). This showed that there cannot be such a complete axiomatization of second-order logic as there was for first order logic. What became known in the case of first order logic as Gödel’s Completeness Theorem simply cannot hold for second-order logic.