Why comparison sorts need n log n time

Someone claims a general comparison-based sorting algorithm can run in $O(n)$ time. How do you show this is impossible?

A decision-tree counting argument. Each comparison has two outcomes, so $k$ comparisons distinguish at most $2^k$ cases, but the algorithm must tell apart all $n!$ input orderings. Hence $k\ge\log_2 n!=\Theta(n\log n)$ in the worst case. Linear time is only possible with extra structure, such as counting or radix sort on bounded integer keys.