The big difference between ethical and aesthetic axioms and the axioms of empirical science is that the latter are so widely accepted that they are not even recognised as axioms, for the most part. If I say "water boils at 100 degrees celcius", this can be proved or disproved to the satisfaction of just about anyone by measuring the temperature of boiling water on several different occasions with several different thermometers. The means of verification contains as it were "hidden" axioms: that checking the boiling point several times with different equipment and obtaining consistent results allows one to generalise about the boiling point of a substance under certain conditions. One could go a level deeper and point out the (axiomatic) assumption that a physical law proved here and now applies to all time and space, the assumption that a logical deduction applies to all possible universes, the "axioms" of logic itself, including rules for using the term "axiom", definition of "rule", definition of "definition"... Fortunately, we hardly ever have to go to such lengths in scientific fields because everyone agrees on the basic axioms. Now that I think of it, this could be used to define a field as a science: a field is a scientific field when the underlying axioms are well-defined and not in dispute by the scholars in that field.
This all stands in stark contrast to ethics and aesthetics, where axiomatic statements (defined as statements taken as given, not dependent on any more basic assumptions) are in dispute all the time. For the record, I am all in favour of being nice to people, opposed to torture and murder, etc. I take these as "axiomatic", meaning that I cannot give a more basic reason behind my acceptance of these beliefs. Some philosophers may push the axiom one level lower, and say, for example, "murder is wrong _because_ it decreases the net happiness in the world". In that case, the axiom is the utilitarian belief that "the good is the greatest happiness of the greatest number". However - and this is the point of this extended reply - there are many who would reject these axioms, especially if they are not of a liberal democratic bent, and there is no way to argue against them as being "irrrational" because if the axiom were rational or irrational it wouldn't be an axiom! If an advanced alien species decided to wipe us out because they regard us in the same way as we regard bacteria, do you seriously think you have a chance of convincing them they are doing something "evil"? What will your argument be when they point out the clause in the Handbook of Intergalactic Ethics which says (after the preamble where it says "we hold these truths to be self-evident") "...more advanced species have the right to enslave, consume or destroy less advanced species." It isn't the same as if they got the boiling point of H2O wrong, is it?
It is perhaps not as easy to get the H2O boiling point right, but you did not convince me of any fundamental impossibility of scientific ethic. Now, I believe that if there is any scientific ethic
then it cannot be normative and cannot give moral injunctions. It will only give theories, which can be applied to solve problems like "if you agree with this and that principle of ethic then you should agree with such and such other principles". You tell me that there will be too much theories. But I can imagine that some will win, or that some will be deduced from other more basic principles.
I don' t really believe that there are scientific fields, I believe there exists scientific attitude which consists in trying to be the clearest possible with oneself and the other, and I believe that is valuable relatively to any question pertaining either in the "human" or "exact" fields, which are today artificially separated (making both of them less exact and less humane).
Anyone having some understanding of science know that it goes from the doubt to more doubts, from astonishment to more astonishment, from vanity to modesty; so that reason can only make bigger our possibilities, falsify our reductions and pinpoint toward the vastness of our ignorance; so that (and that's a theorem with comp) scientific attitude is ethical by itself by making you *ever* more modest, and cautious.
It is only a reductionist view of science which is in contradiction with the idea of scientific theology.
The interest of Godel's theorem in this setting is that it demolishes in one strike a giant class of reductive views on "just" the world of numbers and machine. In the same vain deontic logic exemplifies the natural complexity and intractability of ethical questions: actually in any open rich domain science just gives negative hints of the type: it is not that, neither this, nor .... With comp there could even be some metatheorem like "any normative ethical theory" is unethical, etc.