Conflating syntactic unprovability with semantic model-theoretic multiplicity commits a use-mention error: G_F being unprovable in F does not entail F lacks a unique intended interpretation, only that F cannot prove it.
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An interpretation for an applied modal language comprising the actual and merely possible individuals and worlds the language is understood to be about
model-theoretic(mathematical logic)
Related to the study of how logical symbols and formulas correspond to real structures and meanings (rather than just abstract rules).
semantic(describing the level of word meaning)
Relating to the meaning of words and sentences.
use-mention error(fundamental logic error in arguments)
A logical mistake where you confuse using a word with talking about the word itself—like saying 'dog has three letters' when you meant the word 'dog' has three letters.