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- Given a Structure , is the Set of True Sentences (which I think is called the full theory generated by the Structure) with, say, countable symbols. Is there always a bijection between the true sentences (Semantic ) vs the Theorems (Syntactic)?

Given a Structure , is the Set of True Sentences (which I think is called the full theory generated by the Structure) with, say, countable symbols. Is there always a bijection between the true sentences (Semantic ) vs the Theorems (Syntactic)? I believe this depends on the existence of a model, which itself depends on the consistency of the theory. Sorry, I am a bit rusty with these concepts I last saw some 10 years back . Is it reasonable to assume/define the theory as the maximally-consistent set of wffs? If so, by one of Godel's theorems, a model will exist. If there are finitely-many symbols in our language, there are countably-infinitely many wffs. Then these will be mapped into truths in the/a model. Am I on the right track here? Just need a little push before getting into re-reading the definitions more carefully.