Deliberate mistake: When Ryan is entering the Pentagon to use the hotline, he swipes Cabot's ID card multiple times until finally the gate opens. There are two problems with this. First, if the system beeps, as it did every time he swiped, that means the system saw the response from the card so he shouldn't have needed to keep swiping. Second, the entry to the Pentagon uses a two-factor system, meaning you need to swipe and then enter a corresponding PIN on the keypad that was quite visible in the scene. This is to prevent exactly what Ryan did, namely steal an ID card and use it to get into the Pentagon since the thief wouldn't know the PIN. Since the keypad was there, for the movie the director must have decided all the real steps were too hard to fit the kind of scene he wanted so left it out. (01:44:00)
Deliberate mistake: When asked about the effects of a gas attack on a village in Russia, the president is told that the effects were similar to end stage cerebral palsy. One major problem - cerebral palsy is a chronic condition, not acute. The condition itself is not fatal.
Answer: The song is the classic "Nessun dorma", composed by Puccini and sung (in this version) by Bruce Sledge.