TITLE: Who first explicitly noted that second-order logic is unaxiomatizable?
QUESTION [10 upvotes]: As every student now knows, second-order logical consequence is unaxiomatizable. (At least when we read the second-order quantifiers in the natural way, as running over all possible properties on the first-order domain).
Does anyone happen to know who, back in the glory days, was first really clear and explicit about this?

REPLY [5 votes]: Leon Henkin stated this fact without reference in his 1950 paper in the JSL where he proved the completeness theorem for second-order logic in Henkin semantics [1].  
1: http://www.jstor.org/stable/2266967