In the fourth chapter of "Counterfactuals", David Lewis tries to justify his positions about modal realism. He claims that: "We might take them [modalities] as metalinguistic predicates analyzable in terms of consistency: 'Possibly P' means that P is a consistent sentence [...] If a consistent sentence is one whose denial is not a theorem of some specified deductive system, then the theory is incorrect: no falsehood of arithmetic is possibly true, but for any deductive system you care to specify either there are falsehoods among its theorems or there is some falsehood of arithmetic whose denial is not among its theorem"
"We might take [modalities] as quantifiers over possible worlds where these are treated as linguistic entities: maximal consistent sets of sentences (atomic sentences of sentences of a language enriched by the addition of names for all the things there are, that is diagrammed models) [...] But again the theory would be circular or incorrect, according as we explain consistency in modal terms or in deductive (or purely model-theoretic) terms"
Two things are not so clear to me: why for any deductive system we specify we either have false sentences in its theorems or there is some falsehood of arithmetic whose denial is not among its theorem?
Second, why would the theory be circular or incorrect if we were to consider possible worlds as sets of consistent sentences or set of diagrammed models?