I must say I'm slightly irked by yet another vague and unfitting application of Godel's incompleteness theorems (GIT). The first one roughly states that given a consistent system F there will be some sentence GF such that you cannot derive in F that GF or not(GF). You can then add GF as an axiom to F resulting in F' but then you will be able to construct GF', and so on. This GF you can construct is called the Godel sentence of F.
If you were to encode some rules of ethics in a formal system E you could interpret it such that X is ethical (or not) if you can derive some sentence S (or not(S)) and that sentence S represents X. E will not decide on everything indeed if E is consistent (thus never deriving both some S and not(S)) and capable of basic arithmetic (what does that mean in terms of ethics?!). However, to conclude anything non-trivial from this is dangerous. First of all everyone then has to interpret what the sentences in E mean in an universal way. Then if you construct a GE does that sentence even mean anything intuitively? Will there even be any other sentence not derived from GE that is not decided upon? It is likely that E, while not deciding on everything, decides on much more than any persons ethical intuition. Surely if E is non-trivial there are some complicated sentences that it decides on that no person could interpret. And don't forget the arithmetic issue. If you just include arithmetic in E then all undecided sentences may simply be sentences of arithmetic. If the ethics part of E and arithmetic part are one whole, what would that even look like?
GIT is quite technical and applying it is troublesome. I have only read a bit of actual literature on the topic and even with my limited knowledge of it I can tell that the applications I see in the wild are shaky at best.
there”s no possible set of rules that will completely coincide with our intuitive sense of good and bad behavior in all potential scenarios.
If you just want to come to this conclusion, all you need is a counter example. Take some system of ethics E. If there is any situation such that some person A asserts S and some person B asserts not(S) then either E only decides on at most one making E not universal or E decides on both making E inconsistent. Basically if there is any disagreement on ethics (which there certainly is) you simply cannot construct a consistent and universal set of rules. We really don't need to open Pandora's box and get GIT involved.
Sorry for this wall of text. I'm not angry and I don't want to start a fight lol. I just think that people are a little too eager to apply GIT to all kinds of things. A little pet peeve of mine I guess.
