This is the problem with the digital world. Time is analogue and continuous, digital clocks are just a quantised approximation.

Midday and midnight are points in time that have no duration -- as soon as they are observed they are passed, so midnight is 12am and midday is 12pm.

This is easier to visualise on an analogue clock with a continuous seconds hand. Although the hand sweeps past 12 it spends no time there.

Alternatively think of a digital clock with very high precision. While your ordinary clock will show 12:00pm at midday for a full minute, your high precision clock might be showing 12:00:00.0000001 -- indisputably "post meridiem".