In military operational writing the problem is avoided by never using 0000 or 2400. Only 0001 or 2359 are permitted.
I used to work for a company that did setups for large corporate events. It was not uncommon for us to get access to the facility starting at midnight. Our crews would then work all night to set up a large breakfast or luncheon event.
Our crew call sheets ALWAYS gave the call time as (Day of Week, 12:01AM). It kept people from showing up on the wrong day.
Thank god for you GESancMan. I finally feel understood 
Here’s a more interesting question. To me, Tuesday night lasts until dawn in Wednesday morning. Wednesday morning doesn’t start until dawn.
Because you are just looking at the hour rather than at the minutes. 12:01 PM is one minute post meridian. 1:00 PM is one hour post meridian. If you are using an ante/post meridian system, then it makes no sense to start it at 1 hour past the meridian rather than one minute (or one second).
If you’re using the 24 hour clock, then there is an hour 0. Anyway, isn’t noon really “12 m”?
Yeah it would seem to make more sense if 6am/pm were the dividing lines. Start off the day in the fields/farm at 6am, return at 6pm. Better yet, I wish we’d just switch to military time, but despite what I was told in grade school, that’s a pipe dream, just like converting to the metric system 
In Thai hotels, it’s common for English signs at the front desks to say, “Check-out time, 12:00 AM.” They mean noon, of course.
At the English-language newspapers I worked at there, we were instructed never to use 12am or 12pm but rather midnight and noon.
Yes, yes it is.
One can also think of it as “the first hour” post meridian, which to me makes as much, if not more sense, than saying 12:05 is five minutes past meridian, rather than 12 hours 5 minutes post meridian or something like that.
Look, I understand what a.m. and p.m. mean. It’s a fair enough question to ask why one minute post meridian is 12:01? Doesn’t that seem odd to you at all?
To people arguing there is obviously no ambiguity, the Wikipedia table linked to above gives plenty of examples with midday designated 12 a.m., on one hand (e.g, by the US Government Printing Office in 2000), and with it designated 12 p.m. on the other (e.g., by the same US GPO in 2008!) So there is plenty of ambiguity. Not to mention the confusion evinced in this thread!
BTW any reasonable calendar obviously has a year zero: for instance 0 a.d. = 1 b.c, -1 a.d. = 2 b.c., etc. (Note this Christian calendar actually started up after 500 A.D. anyway.) To keep the number positive, astronomers use Julian days. Which start at noon.
As I mentioned, noon used to be 0h astronomical time. But the civil day begins at midnight, so noon is 12h. Therefore one minute past noon must be 12:01.
(If you are up during the wee hours of the night, is it late, or just really early? Past midnight you are closer to noon, so it must be a.m.)
ETA 12 and 0 are congruent modulo 12… So 12:01 p.m. = 0:01 p.m. I see your point about wanting a minute after noon to be 0:01 p.m., but it seems this convention was never used except by the aforementioned astronomers who were using a 24-hour clock anyway.
I see the OP’s point that it seems to make little sense that the label 12 is the beginning of the morning or the beginning of the afternoon. But it would make even less sense to have the morning of afternoon start at 1:00. When you’re ten minutes into the morning, the time should logically be zero hours and 10 minutes, not 1 hour and ten minutes. So if you just think of the time that we label 12:XX as indicating zero hours and XX minutes (as DPRK says just above), it does make sense.
Well, 2400 makes about as much sense as saying it’s 13 o’clock (civilian time).
And if you have enough functioning brain cells to understand what 0001 is then understanding clearly what 0000 is should hardly be an issue.
When I become Queen, I’m changing 12 to 00. I was always a whiz at those ‘what comes next in the sequence’ tests, so the fact that 12AM & PM deviate from the pattern just grinds my gears.
While we’re at it with the hypothetical nanosecond clock: When it’s daytime and the clock reads 12:00:00.000000001, that’s after noon, right? But now think: Most clocks are not nanosecond clocks. At that exact moment (which we agree is after noon), what would an ordinary, 1-second clock be reading?
It really isn’t 'much of the world".
Actually, it’s only the British & former British colonies that still use AM/PM times. Nearly everyone else uses 24-hour clocks now. Of course, that includes the UK & USA, rather important countries in the economy.
They overlap for me. 4:00 AM is Tuesday night if I’m still up from Tuesday, but Wednesday morning if I have to get up early.
The starting date/time and ending date/time of my auto insurance policy (and I presume ALL auto insurance policies) are shown similarly.