Many people see themselves in various groups (member of the population of their home country, or their social network), and feel justified in caring more about the well-being of people in this group than about that of others. They will argue with reciprocity: "Those people pay taxes in our country, they are entitled to more support from 'us' than others!" My question is: Is this inconsistent with some rationality axioms that seem obvious? What often-adopted or reasonable axioms are there that make this inconsistent?
Well, nobody can dictate which terminal values you should have, i.e. the utility function is not up for grabs. However, if you choose a class of things similar to you
(e.g. everything which weighs 150 lbs, everyone of your race, every human less that 5km away, all human brains which weigh more than 1 pound, everything which has human DNA or possibly everything living)
, then you can limit your utilitarianism to these things and be a perfect utilitarian w.r.t. this group. I guess she meant it this way, although I would embrace clarification or confirmation.
The most important case is where your can't yourself arbitrarily declare your own values.