"Parfit (1984) rejected Totalism and Averagism...[and] failed to find any alternative axiology that he himself considered satisfactory, but held out hope that...some fully satisfactory population axiology, called 'Theory X' by way of placeholder, might be found. Much of the subsequent literature has consisted of attempts to formulate such a 'Theory X'. However...every extant population axiology is open to serious objection: if it does not entail the Repugnant Conclusion then it entails the Sadistic Conclusion, or is anti-egalitarian, or has obviously unacceptable implications concerning future-people cases, or otherwise leads to some similarly serious objection. If this were where matters rested, one might yet hold out hope that the fully satisfactory 'Theory X' is lurking just around the corner. However, several authors have also formulated impossibility theorems for population axiology. These are formal results that purport to show, for various combinations of intuitively compelling desiderata ('avoid the Repugnant Conclusion', 'avoid the Sadistic Conclusion', 'respect Non-Anti-Egalitarianism', and so forth), that the desiderata are in fact mutually inconsistent: that is, simply as a matter of logic, no population axiology can simultaneously satisfy all of those desiderata"