If a formal system has a single statement that is simultaneously true and not true, then you can prove any statement (and its opposite) in that system, and it is therefore useless. This was known before Gödel. His insight was that in a system that is not inconsistent (and that is complex enough to represent arithmetic), there will be some situations where given some proposition x, you can neither prove x nor ~x. That’s not “simultaneously true and not true” (“true” is in the territory, a formal system is the map), it just means the truth value is unknowable within the system.
In any case, I think this is fairly irrelevant to moral philosophy, because Gödel’s theorems are about formal systems representing number theory. I suppose you could somehow represent empirical statements (including moral statements, if we agreed on exactly what facts about reality they signify) in that form — take a structure representing the entire universe as an axiom, and deduce theorems from there — but that’s rather impractical for obvious reasons, and there’s nothing that really suggests that this provides any analogous insights about simpler and more possible modes of reasoning. In fact, you could change your statement to talk about any area of knowledge (say, “If science is encapsulated by a formal system...” “If aesthetics is encapsulated...”) and it would make just as much sense (or just as litte, rather).
If a formal system has a single statement that is simultaneously true and not true, then you can prove any statement (and its opposite) in that system, and it is therefore useless. This was known before Gödel. His insight was that in a system that is not inconsistent (and that is complex enough to represent arithmetic), there will be some situations where given some proposition x, you can neither prove x nor ~x. That’s not “simultaneously true and not true” (“true” is in the territory, a formal system is the map), it just means the truth value is unknowable within the system.
In any case, I think this is fairly irrelevant to moral philosophy, because Gödel’s theorems are about formal systems representing number theory. I suppose you could somehow represent empirical statements (including moral statements, if we agreed on exactly what facts about reality they signify) in that form — take a structure representing the entire universe as an axiom, and deduce theorems from there — but that’s rather impractical for obvious reasons, and there’s nothing that really suggests that this provides any analogous insights about simpler and more possible modes of reasoning. In fact, you could change your statement to talk about any area of knowledge (say, “If science is encapsulated by a formal system...” “If aesthetics is encapsulated...”) and it would make just as much sense (or just as litte, rather).