The soul is immortal is the belief that hardly goes along with the modern construction of reality. The soul is ignored by the modern day theories of existence though Nietzsche has propounded the the theory of eternal recurrence. Nietzsche has failed to convince the modern day philosophical schools of thought that soul exists and exhibits the pattern of birth and death alternating each other. What could be the reason behind turning a blind eye to Nietzsche? May be it has to do with the pursuance of Nihilism during the two world wars and hence it fell away with the succeeding generation of Philosophers.
Even Pythagoras after theorizing with the hypotenuse as the square of sum of squares of adjacent sides fell unconvincing with his ideas of immortality and transmigration of souls. We could believe the mathematical abstractions but the other abstractions are ignored as there is not enough proof of them. The abstraction that soul exists comes from the way we abstract things that are themselves pure abstractions like mathematical figures squares. circles, rectangles etc, yet the abstraction of the soul are not believed in.
Now we abstract time from the present, from the nowness of moment thus we abstract moments after moments of time. Such moments come from fictional substance which unfolds all that we could abstract and in which the time exists, the fictional substance could mean the substance through which time propagates. The abstraction regarding the immortality of soul also comes from the moments we abstract from time. If each abstracted moment is true, the possibility of soul is also true. Nature is part memory and part consciousness. All that consciousness encompasses is stored in the nowness of time, the current moment of which I am made aware by neurons transmitting in central nervous system. From the consciousness I feel reality but the unconscious stands for the parts of psyche that are unknown and may relate to our previous births.
Thus we could safely assume the possibility of the soul exists.