That the clock starts at 12 has annoyed me for quite a long time (i.e., 12PM comes before 1PM). I kind of wish that non-military usage also used 0:00-11:59, if only for the logical consistency of it.
Alternatively we could just switch 12PM and 12AM, but that would mess with midnight == start of the day.
Unfortunately I think I'm doomed to suffer through this inconsistency, since it doesn't seem to bother anyone else.