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).
On Wei_Dai's complexity of values post, Toby Ord writes:
The kind of moral realist positions that apply Occam's razor to moral beliefs are a lot more extreme than most philosophers in the cited survey would sign up to, methinks. One such position that I used to have some degree of belief in is:
Strong Moral Realism: All (or perhaps just almost all) beings, human, alien or AI, when given sufficient computing power and the ability to learn science and get an accurate map-territory morphism, will agree on what physical state the universe ought to be transformed into, and therefore they will assist you in transforming it into this state.
But most modern philosophers who call themselves "realists" don't mean anything nearly this strong. They mean that that there are moral "facts", for varying definitions of "fact" that typically fade away into meaninglessness on closer examination, and actually make the same empirical predictions as antirealism.
Suppose you take up Eliezer's "realist" position. Arrangements of spacetime, matter and energy can be "good" in the sense that Eliezer has a "long-list" style definition of goodness up his sleeve, one that decides even contested object-level moral questions like whether abortion should be allowed or not, and then tests any arrangement of spacetime, matter and energy and notes to what extent it fits the criteria in Eliezer's long list, and then decrees goodness or not (possibly with a scalar rather than binary value).
This kind of "moral realism" behaves, to all extents and purposes, like antirealism.
I might compare the situation to Eliezer's blegg post: it may be that moral philosophers have a mental category for "fact" that seems to be allowed to have a value even once all of the empirically grounded surrounding concepts have been fixed. These might be concepts such as "would aliens also think this thing?", "Can it be discovered by an independent agent who hasn't communicated with you?", "Do we apply Occam's razor?", etc.
Moral beliefs might work better when they have a Grand Badge Of Authority attached to them. Once all the empirically falsifiable candidates for the Grand Badge Of Authority have been falsified, the only one left is the ungrounded category marker itself, and some people like to stick this on their object level morals and call themselves "realists".
Personally, I prefer to call a spade a spade, but I don't want to get into an argument about the value of an ungrounded category marker. Suffice it to say that for any practical matter, the only parts of the map we should argue about are parts that map-onto a part of the territory.