The argument assumes the standard fine-structural account of L, but Sy Friedman's inner model program shows enriched L-like models can accommodate large cardinal structure while retaining constructibility-style definability.

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Constructibility-style definability(as used in mathematical logic): The quality of being able to describe or build something using only simple, step-by-step logical rules, similar to how Gödel's constructible universe is built.
Fine-structural account(as used in mathematical logic): A detailed technical description of how mathematical structures work, breaking them down into their smallest components and rules.
Inner model program(as used in set theory): A research project in logic aimed at understanding and constructing different mathematical universes that sit inside larger mathematical frameworks.
L (the constructible universe)(as used in set theory and logic): A theoretical mathematical universe built from the simplest elements using only basic logical operations, imagined by logician Kurt Gödel as a way to study what can be mathematically proven.

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