Quote from Gringinho:
See model theory - the implications of Incompleteness is that no logical system can contain/model "everything".
Fair enough, the implication I see arising from the incompleteness theorem is that certain models will have to account for axioms that cannot be proven true or false.
The problem is that this accountability can only be provided by incorporating intuition. The problem with intuition is that it can only be grasped, it cannot be used to demonstrate truth in any absolute sense. Nor do I see any way to prove absolutely that intuition can be used to grasp anything at all.
If a person does have an intuitive grasp of a model it can however be used to solve any amount of hypothesis which make up the underlying parts of the complete model. This is for example why Plato considered his intuitive grasp of 'the Good' to be above any form of hypothesis.
How this concerns ethics is the problem proposed by G.E. Moore that states that what is 'good' cannot be defined. Doing so invariably creates a tautology which devoids it of meaning. The result is that no ethical axioms can be discovered which are objectively true or false.
If good is defined as say, acting on behalf of general happiness the open question "what is good?" can no longer be asked since the answer is presupposed to be "acting on behalf of general happiness". Moore's own proposition to tackle this problem is that ethical propositions are either true or false, but a verdict can only be reached along intuitive lines.
How this relates to objectivism is that as a philosophy it makes a strong appeal to a person's intuition in order to prove the truth of its axioms. For example, the mere fact that Greenspan makes an argument does not prove conclusively that he exists. As a counter argument he could have stated that his utterance of some argument was just an illusion and he doesn't exist after all.
"Greenspan exists" does however provide a person with an intuitive axioma that's strong enough to base a philosophical system on. Which, by the way, objectivism does. It starts where it should start, with ontology in order to derive further conclusions about ethics and politics.