Ariadne, the daughter of King Minos in Greek mythology, provided the thread Theseus used to escape from the labyrinth because she was his lover. (The labyrinth itself was designed by Daedalus.) She escaped Crete with him but then Theseus abandoned her on the island of Naxos.
The movie Ariadne designs the labyrinth, which is not even shown. She provides the entrance to the heart of the labyrinth. She is not the lover, nor does she escape with him, nor is she a princess. Yet, the name is still a blatant symbolic reference. If it doesn't have any meaning, it's not just hammering away like the score, it's incompetent.
How does Page help DiCaprio escape?
We cannot plausibly read Page as helping DiCaprio escape a labyrinth of dreams via inception (the purported plot for inception against Murphy being the disguise.) That would involve taking the internal rules of extraction and inception seriously.
Questions to answer if DiCaprio is the target of an inception
When was DiCaprio put under?
If at the beginning of the movie, it's a cheat because prima facie DiCaprio knows how he got there. After that, there is a continuous sequence, but the hallmark of a dream is that it starts in media res. The question does not permit an answer.
Murphy projects Berenger in the second level of dreams, so he must be one dreamer. (Otherwise there is no reason for us to see Watanabe confusing projection-Berenger with Hardy's forgery of Berenger.) Given Page's conspicuously unlikely role in propelling character revelations to the audience and the big save when Murphy is "killed," she must be another. But we can't tell who the other one is (there has to be one for each dream level.) Again, the question does not permit an answer.
Who's a dreamer, a projection or a forgery?
Internal parallelism suggests Cotillard's first appearance is a forgery. Page and Murphy in limbo would be projections while Watanabe would either be a dreamer completing the inception or a projection enabling an apparent suicide. Once again, the question does not permit an answer.
How many levels are there?
By the rules of the game, the events in one level influence the events in the lower dream level. People in the hotel are in free fall when the van is falling. There is no influence from the supposed real world where DiCaprio is put under. We cannot know whether that means the real world is quiet or because we are seeing the real world on screen.
What is the idea being incepted?
That Cotillard is bad seems silly. Cotillard's character name in French is "mal" which means evil. This makes is entirely unnecessary to implant the idea that she is evil. That there is a way back to the US and the kids suffers from there not being any known obstacle keeping DiCaprio away. (Unless the idea is to get him to come back and be arrested?
) That reality is better than dreams, implanted by the Murphy subplot as a sort of reverse psychology, doesn't work because the last level is a dream,
(italics for omitted word!) the same as the first, which means he does not return to reality and there is no resolution.
Something weird doesn't mean it's a dream, no matter what the dialogue says
All the analysis above is kind of tedious. But it seems to somehow be necessary. The implausibility of corporate goons chasing people through the streets of Mombasa or billionaires able to make one phone call and get prime suspects for murder off the hook have inspired some people to think the nonsense that there is another inception game going on. These are routines premises for all sorts of movies.
It's all a dream? An aside.
Some have gone beyond that, to just saying it's all a dream. Psychologically, this kind of dream is nonsense. The movie is done like virtual reality, not like a dream. This kind of elaborate story dream is just not the kind of wish fulfilment people dream. Even more, DiCaprio is not going to dream scenes about Murphy's catharisis. Last, the notion that reality is questionable is not actually a very interesting one. The very notion is a devastating criticism of the movie, not an explication.
What are (some) people thinking?
Some of the stuff that has been proposed is mindbogglingly wrong.
There's a suggestion that Cotillard being on a ledge across from DiCaprio's window is nonsense and therefore a dream metaphor for their separation. In fact, Cotillard has to be out of his reach or she could not jump at all.
It is plain that Caine does not have custody of the children in Paris or DiCaprio would be visiting them. The conclusion that his appearance in the US airport means DiCaprio is dreaming is absurd, since Caine is not nailed down in Paris. Flying to the US for the family reunion (or the arrest of the son-in-law, if I haven't confused the Caine/DiCaprio relationship,) is perfectly natural.
The reason the top in the final scene spins so long is for "suspense," because the top is starting to wobble, which means it's going to fall offscreen. There isn't any doubt that it will fall,
whatever debate about whether it means anything.
Page as Ariadne is so perversely wrong as to suggest irony, except there isn't the slightest reason to think Nolan can write so subtly. It appears, from his previous work, that Nolan has the peculiar idea that ambiguity is deep. Well, it isn't, and stuff like this proves it.
Inception's about moviemaking!
Lastly, it has been suggested that it's all purportedly a dream, but is really a meditation on moviemaking. The only plot points that explore this is Murphy's imagined reconciliation with his father contrasted with DiCaprio's rejection of his wife. The second is not convincingly handled. Worse, we are not given time to wonder about how we feel about Murphy's reconciliation. Are we pleased at a happy ending? Are we angry because he has been abused on a profoundly deep emotional level? Burying a climactic scene like that is deliberately obscuring the point, unless it wasn't really the point. A movie about the attractions of moviemaking that forces us to endure so many boring action scenes is shockingly inept. If the point is just to mess with people's minds, this is no more enlightening than a joy buzzer.
It's not good, it's original!
This movie is amazingly original but it is not deep. It is about whether DiCaprio wins, a vicarious wish fulfilment story. The other characters are poorly sketched or blatant nonsense. The SF premises are entirely unexplored. The only real exploration of the dreams (or movies) vs. reality theme is the Murphy catharsis which is at best a minor part of the movie. Maybe it should have been a climactic moment with huge thematic resonance, instead of just a human moment for which we poor viewers were grateful, but it wasn't. If it wasn't for the originality of the concept and the free fall sequence, the movie would have to be rated poor!
(extensive editing for clarity, or so I hope)