First off, I would find it highly unlikely that the only habitable place after a nuclear war would be a portion of Manhattan. I would think that Manhattan, being the third largest borough in New York at approximately 23 square miles and very densely populated, would be one of the FIRST places targeted by a foreign power (or terrorist group) to nuke off the face of the earth. That to me is illogical scenerio #1. How could that one itsy-bitsy spot on the planet escape all harm????
Secondly, I agree with the subsequent illogical nature of the story without explanation as to how and where this advanced technology was manufactured along with the production of enough food and water to sustain whatever population was fortunate enough to have been protected by the magical "anti-catastophe bubble" they must have had over Manhattan when the proverbial sh** hit the fan.
I pick movies apart too, especially when the story line defies possibility far beyond all reason or extreme stretch of the imagination. At least make the science and fiction reasonably plausible.