Under Henkin semantics, the argument that only countably many structures are characterizable fails, because the expressive power of second-order logic collapses to that of many-sorted first-order logic, which characterizes structures up to elementary equivalence rather than isomorphism.

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Characterizable(as used in logic): Able to be uniquely described or defined using the rules and symbols of a logical system.
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.
Many-sorted first-order logic(as used in mathematical logic): A version of basic logic that organizes objects into different categories or 'sorts' (like separating numbers from colors), making it more flexible than traditional first-order logic.
Second-order logic(as used in mathematical logic): A formal system that goes beyond basic logic by allowing you to quantify over (talk about) properties and relations themselves, not just individual objects.

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