The one,which is born lives and dies within the world is not the "self" because the self is ever formless.

The birth, life and death are part of the waking experience. Waking experience is mere  illusion  standpoint of the  formless  soul or consciousness, the innermost self .  It is no use of saying that, we are not born, and we do not die. Whatever is born will die. One has to understand the fact that the birth, life and death takes place in the illusion, which comes and goes as waking experience.  The formless witness of the three states is real, which is our true identity has no birth and has no death. The one,which is born lives and dies within the world  is not the "self" because the self is ever  formless.

Thus, it is wrong to judge and say we are not born, and we do not die by limiting the self to the waking entity or physical entity or false self or ego. The true self is the soul, which is the birth less and deathless eternal identity.

  If one limits the self to the physical entity alone, one gets only half-truth.  The self pervades all the three states, as its formless substance and witness.  The formless substance and witness, which is the true self is the soul. Without knowing the true self-mere saying, we are not born, we do not die is not the knowledge, which has come from depth, it is mere bookish knowledge. Unity in diversity is not possible through bookish knowledge.

 Sri,Sankara says:  The world is unreal . He never said body is unreal. He would have said only birth and death is   unreal, he did not say that, he said the world is unreal.   He meant   the world including the body, because the body and world appear together and disappear together [waking/dream].  Then we have to think, what remain without the body and the world as reality, which is Brahman.

How can the birth, life, death, rebirth, and reincarnation, which take place in unreal world [waking/dream], can be true.   The one, which is aware of the birth, life death, rebirth and reincarnation [illusion/unreal], is formless reality, which is the true self/soul.  The formless self is true identity, which is birthless and deathless. The true self is the formless witness of the birth, life and death and rebirth [illusion/unreal].