It wouldn't be set right after the end. And not everything would be perfect. The nations would probably still quarrel with each other. There'd be wars and misunderstandings. There would be a lot we could do. 

Anyways there's a really cool website that gives the timeline of the world from the "first age". 
http://www.sevenspokes.com/chronology/3rd-fy.html
I'd probably set the story somewhere in these years during the time when Hawking was still alive.