If the metatheory is weakened to predicative or intuitionistic set theory, the Henkin model construction fails because maximal consistent extensions require non-constructive choice principles.
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Non-constructive choice principles(as used in mathematical logic and set theory)
Mathematical tools that let you make selections from infinite collections without having to explicitly describe how you're making each choice; they're powerful but somewhat abstract and 'non-constructive' because you can't always show your work step-by-step.
Predicative(as used in set theory)
A restriction in logic where you can only define something using things that already exist, not by referring to a larger set that includes the thing you're defining.