The Hilbert-Bernays-Löb derivability conditions are themselves extensionally specifiable constraints on syntactic proof predicates, not genuinely semantic or intensional requirements.

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Derivability conditions(mathematical logic): A set of formal rules that describe how provability works in a logical system—basically the 'dos and don'ts' for what counts as a valid proof.
Extensionally specifiable(describing how the conditions can be specified): Something that can be described or defined by listing out all the concrete, actual cases or examples that apply to it, rather than by abstract principles.
Hilbert, Bernays, and Löb(naming the derivability conditions): Three mathematicians and logicians (David Hilbert, Paul Bernays, and Martin Löb) who developed important rules about how formal mathematical proofs work. Their names are attached to specific conditions that describe what counts as a valid proof.
Syntactic proof predicates(what the conditions apply to): Rules or formulas that check whether something is a valid proof by looking only at the symbols and structure of the proof itself—like checking grammar rules without caring about meaning.

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