Arrow Impossibility Theorem
One of the astonishing findings of modern economics (moreover from a student's PhD thesis.) is that a set of quite reasonable-sounding requirements about social choice orderings necessarily implies that there is no method for constructing social preferences from ordinal individual preferences. In other words, there is no rule, such as majority voting (nor any other), for deriving social preferences from arbitrary individual preferences of the kind commonly assumed by economists. The reasonable requirements (axioms) are these:
Completeness: in a choice between alternatives A and B, A is so cially preferred to B, or B is preferred to A, or there is a social indifference between them.
Transitivity: if A is socially preferred to B and B is preferred to C then A is also preferred to C.
Non-dictatorship: social preferences should not depend upon the preferences of only one individual - if every individual prefers A to B then socially A should be preferred to B.
Social preferences should be independent of irrelevant alternatives; i.e. the social preference for option A compared with B should be independent of the preference for C.