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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?

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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

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    Thanks for this. It would be striking indeed if this is the first explicit statement (some 20 years later than you might have expected??).2012-11-22