Your orange takes the zeroth index becoming the first orange: http://i.imm.io/1lWRO.png
People understand the concept of zero perfectly. They know the day doesn't begin at 1am, they know babies aren't born 1 years old. This isn't a computer-specific concept in any way.
This is true, and an interesting point, but also nobody describes a baby as being zero years old, or the time as being zero-o'clock. People avoid zero in these situations and find alternatives - i.e. 3-months or 12.30am.
Personally I see both sides but the point is that whenever you make something divisible, the first unit is 1 and that's what people call 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.
I guess for me personally the perfect solution would be a 12-hour clock that operates on the same principles as the European 24-hour (which I think is the same as the military clock?). So 00:00 to 11:59, twice per day.
But honestly my whole argument is pretty academic, so I'll probably just put up with the clock starting at 12.