Dear Pavel and Dimitry,
thanks for steering me towards your essay! I don't have much time for an in-depth comment right now, but I wanted to point out that it seems your approach is 'Gödelian' in quite another way from many of the other essays in this contest---fellow contestant Hippolyte Dourdent, in his article, quotes the great logician as saying that "time is the means by which God
realized the inconceivable that P and non-P are both true"---a lesson which seems to get to the heart of your approach.
It also seems to me that there is an interesting relationship between your 'areal sets' and the notion of complete sets of commuting observables in quantum mechanics---in the sense that, if a system has a definite value within the observables of one such set, it fails to have a definite value for all others. For instance, an electron having a definite value of spin for one particular direction (and consequently, having a definite value for all observables commuting with the spin-operator in that direction), fails to have a definite value for spin in all other directions.
Also, I quite liked the entry point you chose for your essay---St. Augustine is not somebody you expect to meet in such a contest. But then again, it was him who said about what time is, "if no one asks me, I know; if I wish to explain to him who asks, I know not". Perhaps, still, not entirely inappropriate.
Good luck in the contest!