The Vortex World (ボルテクス界, Borutekusu kai)? is the setting where Shin Megami Tensei III: Nocturne takes place. Unlike the original planet it was born from, all inhabitants walk on the inner layer of the Vortex World and the outside is bare, in essence making it a form of Dyson sphere. In its literal center is the being known as Kagutsuchi, who radiates energy that affects all inhabitants of the Vortex World.
Its existence is the result of Tokyo being made into a circular world through an apocalyptic event known as the Conception. Everyone outside of the Shinjuku Medical Center was killed instantly and turned into wandering spirits, Manikins or Magatsuhi, as Tokyo was uprooted and twisted into its spherical shape by Kagutsuchi, the pulsating moon-like being who shines brightly at the center of this world.
When someone with both Reason and a demonic sponsor proves their ideals worthy by besting the opposing forces, gaining the influence of the Demi-Fiend, and by defeating Kagutsuchi as a final test of the Reason-holder's strength, Kagutsuchi will release his powers and reshape the world based on the ideals of the chosen individual.
In the First Kalpa, the Lady in Black reveals that the Vortex World is one of countless worlds that experiences rebirth in the Amala Universe. Whether the world flourishes or stagnates, all worlds are subject to death and reincarnation after a certain period of time has passed.
In more specific terms, The Vortex World is just one of many worlds in the extremely-if-not-infinitely vast Amala Universe. There are other worlds like it being reborn through the power of creation. In addition to these worlds, there are others in which demons and gods reside, worlds barren of the power of Magatsuhi. They are branded 'nonexistent' due to their lack of power and the true death of their home worlds. They wish to become "real" by the power of Creation and the Conception, and to that end they seek out worlds with such power in the endless flow of Amala. In the Vortex World and others with the power of creation, these beings can do very little, since their very existence is forbidden, and wish to change that fact.