The Ascians are trying to rejoin all of the shards with the source, which would result in humanity being returned to how it was originally and Zodiark becoming whole again.

What I don't quite understand is... what about the 13th shard? Emet Selch admits that they screwed up, causing the 13th shard to turn into void.
So with one of the shards lost, is it even possible for the Ascians to reach their goal?