A good chunk of us who have just seen Inception have started debating over whether Leonardo DiCaprio’s character Cobb is still stuck in the dream world. The film’s final shot is his totem spinning on the table and before we see if it topples (a sign of being in the real world), it cuts to black.
Throughout the film we regularly see flashes of Cobb’s children as he remembers them right before he goes on the run. His young daughter appears in a light red dress and his son in some plaid shirt. Now, if we are to apply the A Beautiful Mind theorem, we might be able to deduce whether Cobb is awake or dreaming.
In A Beautiful Mind, John Nash is able to discern between illusion and reality by recognizing the ageless nature of his illusions. As he ages, the characters he hallucinates stay the same age, which therefore proves to himself that he’s just imagining these characters.
Now, we can’t exactly apply the same logic to Inception because we don’t know exactly how long Cobb has been exiled from the U.S. From the way he talks to his children on the phone in the film and his discussion with Miles (Michael Caine) about his occupation shift, it could be up to a year or two. (I’m assuming he’s been a dream thief for a while; Cobb comments that there aren’t a lot of legitimate jobs for a man running from the law, and it seems that establishing oneself as a dream thief would take some time). Since we don’t know the exact length of time between his wife’s suicide and the time of the film’s events, we can’t determine if his children should have aged more when he sees them at the end of the film.
However. When he comes home and goes to his children, they are wearing the exact same clothes as those in his memory. Before he leaves the country, he sees his children in red and plaid, playing outside in the grass. When he comes home, they are outside, playing in the grass, in the same clothes. If Cobb has been gone for any length of time, the children would look different in some way, if not in age, then at least in clothing. The fact that Nolan had the children wearing different clothes in a separate memory (the one where he sees his kids and Mal on the beach) could be a clue that if Cobb were experiencing something new, the kids would have been wearing clothes not connected to his memories.
So there you have it. The A Beautiful Mind theorem and what I believe to be the true ending of the film: Cobb’s still stuck in his subconsciousness.