1 Eternal recurrence involves reference - note, merely reference - to personal identity as 'The Gay Science makes clear :
The heaviest burden.? What if a demon crept after you one day or night in your
loneliest solitude and said to you: "This life, as you live it now and have lived it, you will have to live again and again, times without number; and there will nothing new in it, but every pain and every joy and every thought and every sigh and all the unspeakably small and great in your life must return to you, and every thing in the same series and sequence?and in the same way this spider and this moonlight among the trees, and in the same way this moment and I myself. The eternal hourglass of existence will be turned again and again?and you with it, you dust of dust!"?Would you not throw yourself down and gnash your teeth and curse the demon who thus spoke? Or have you experienced a tremendous
moment in which you would have answered him: "You are a god and never did
I hear anything more divine!" If this thought gained power over you it would, as
you are now, transform and perhaps crush you; the question in all and everything "do you want this again and again, times without number?" would lie as the heaviest burden upon all your actions. Or how well disposed towards yourself and
towards life would you have to become to have no greater desire than for this ultimate eternal sanction and seal?' ('The Gay Science, tr. W. Kaufman, NY : Random House, 1974, 273-4)
2 The most natural reading yields a purely ethical interpretation, however, involving no physical recurrence of persons. On these lines, the doctrine means: 'Act as though you had to relive your life innumerable times and will relive it innumerable times' (Pierre Klossowski, 'Nietzsche and the Vicious Circle', tr. DW Smith, London : Athlone Press, 1997, 56-7). 'Act as though you have to relive' does not imply 'Act in light of the fact that you have to relive'.
3 We might also wonder how generally the ethical interpretation is to be taken. For whom is it meant ? In 'Ecce Homo' he says :
'The basic idea of the work [Zarathustra], the thought of eternal return, the highest formula of affirmation -, belongs to August of the year 1881 : it was thrown onto paper with the title '6,000 feet beyond people and time'. That day I went through the woods to the lake of Silvaplana; I stopped near Surlei by a huge, pyramidal boulder. That is where this thought came to me' (Nietzsche, 'The Anti-Christ, Ecce Homo, Twilight of the Idols', ed. A. Ridley & J. Norman, Cambridge : Cambridge University Press, 2005, 123).
The 'basic idea' occurs to a solitary individual, Nietzsche, isolated from 'people and time'. A tentative thought is that the basic idea was conceived as being for Nietzsche's sole guidance and not as a message to the world.