A neat idea from Welfare Axiology 

Arrhenius's Impossibility Theorem

You've no doubt heard of the Repugnant Conclusion before. Well let me introduce you to it's older cousin who rides a motorbike and has a steroid addiction. Here are 6 common sense conditions that can't be achieved simultaneously (tweaked for readability). I first encountered this theorem in Yampolskiy's "Uncontrollability of AI"

 Arrhenius's Impossibility Theorem 

Given some rule for assigning a total welfare value to any population, you can't find a way to satisfy all of the first 3 principles whilst avoiding the final 3 conclusions.

1. The Dominance Principle: 
    If every member of population A has better welfare than every member of population B , then A should be superior to B.
   If populations A and B are the same nonzero size and  every member of population A has better welfare than every member of population B, then A should be superior to B
   (Thanks to Donald Hobson for this correction)
2. The Addition Principle:
   Adding more happy people to our population increases it's total value.
3. The Minimal Non-Extreme Priority Principle:
   There exists some number such that adding that number of extremely happy people plus a single slightly sad person is better than adding the same number of slightly happy people. I think of this intuitively as making some amount of people very happy outweighs making a single person slightly sad.  
4. The Repugnant Conclusion:
   Any population with very high levels of happiness is worse than some second larger population of people with very low happiness. 
5. The Sadistic Conclusion:
   It is better to add individuals to the population with negative welfare than positive welfare.
6. The Anti-Egalitarian Conclusion:
   For any perfectly equal population, there is an unequal society of the same size with lower average welfare that is considered better.