The dependence on AC exposed by Cohen's models is a feature of Henkin semantics, where comprehension is restricted, not of second-order logic under its intended, unrestricted interpretation.

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AC (Axiom of Choice)(as used in set theory): A basic assumption in mathematics that allows you to make infinitely many arbitrary choices all at once—something that seems obvious but is actually quite controversial.
Cohen's models(as used in mathematical logic): Mathematical structures created by logician Paul Cohen to show that certain statements cannot be proven true or false using standard mathematical rules.
Comprehension (in logic)(as used in set theory and logic): The principle that you can create a new set by describing what properties its members must have; 'restricted comprehension' means you can't always do this for any description you write down.
Henkin semantics(as used in mathematical logic): A way of interpreting logical languages (created by logician Leon Henkin) that uses a more limited set of possible interpretations than the standard approach, making the logic less powerful but more practical.

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