We show that it is impossible to design a transactional memory system which ensures parallelism, i.e. transactions do not need to synchronize unless they access the same application objects, while ensuring very little consistency, i.e. a consistency condition, called weak adaptive consistency, introduced here and which is weaker than snapshot isolation, processor consistency, and any other consistency condition stronger than them (such as opacity, serializability, causal serializability, etc.), and very little liveness, i.e. that transactions eventually commit if they run solo.