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Infinity for Nietzsche in at least one line of argument involves the eternal return; he refers to it in the *Die fröhliche Wissenschaft* and *Also sprach Zarathustra*; most completely in his *Notes on the Eternal Recurrence*:

Whoever thou mayest be, beloved stranger, whom I meet here for the first time, avail thyself of this happy hour and of the stillness around us, and above us, and let me tell thee something of the thought which has suddenly risen before me like a star which would fain shed down its rays upon thee and every one, as befits the nature of light.

Fellow man! Your whole life, like a sandglass, will always be reversed and will ever run out again, - a long minute of time will elapse until all those conditions out of which you were evolved return in the wheel of the cosmic process. And then you will find every pain and every pleasure, every friend and every enemy, every hope and every error, every blade of grass and every ray of sunshine once more, and the whole fabric of things which make up your life. This ring in which you are but a grain will glitter afresh forever.

And in every one of these cycles of human life there will be one hour where, for the first time one man, and then many, will perceive the mighty thought of the eternal recurrence of all things:- and for mankind this is always the hour of Noon.

This thought is one echoed in Indian Metaphysics - the cyclical universe and in physics via Poincares reoccurence theorem which is traced to a question in Celestial Mechanics - the question of the stability of the Solar System.

But can repetition characterise infinity? Or should it be natality, that is true infinity is characterised by non-repetition that is however 'far out' one goes nothing repeats, there is always some modality, some aspect that is *essentialy* new?

In Spinozoan Metaphysics, for exampe, there are an infinite number of modes that are essentially different from each other; the first two being extension (ie matter) and thought - the incommensurability of the two is exactly the *hard* (ie very dificult and probaby impossible problem of consciousness); here Spinoza is implicitly remarking that the infinity (of God) is characterised by plenitude, by incommensurability and by fullness.

Seems like no. There are an infinite amount of numbers between 0 and 1 without any repetition. – James Kingsbery – 2014-10-27T16:17:44.157

IIt does depend how you view it; take the open unit interval ie all numbers between 0 and 1; and look at it geometrically; then every point locally looks like every other one. Ie repetition. – Mozibur Ullah – 2014-10-27T18:07:47.200

Another way of looking at this is arbitrarily rearrange every point in the interval; does it look different? No, not particularly. The same goes for counting which is labelled 1, 2, 3; but if one sees it as a – Mozibur Ullah – 2014-10-27T18:10:26.657

Sequence of bottles - we have the first bottle, the second bottle, and so on; and every bottle is the same. Ie repetition again. – Mozibur Ullah – 2014-10-27T18:13:26.717

@MoziburUllah Each real number, and each natural number 1, 2, 3, ... is a distinct

setwhen numbers are defined as sets. You agree with that, right? No two are the same. But I agree that geometrically, you have a -- no pun intended -- point. Re the bottle example, did you have in mind the old joke "Aleph-null bottles of beer on the wall ..." – user4894 – 2014-10-27T18:57:09.413@user4894: Sure; but to honest they're all subsumed under the relation of quantity; the point I'm getting at here is that thought and extension are essential different in a way that three and twenty five are not. – Mozibur Ullah – 2014-10-27T19:01:30.963

But every point in the interval has a different distance from zero, so in what way do they 'look the same'? If I took each of them and fed it into a fractal function, the resulting Julia sets and such would be quite different for points very close together. – None – 2014-10-27T19:23:21.120

I said

locallythe same; ie theneighbourhoodof the point; these are fairly basic and easy notions from topology. – Mozibur Ullah – 2014-10-27T19:29:37.433Right, you can abstract away details and make things similar, for tractability. But the detail is there for you to ignore. Ignoring it does not make it less real. You have only finitely much attention, so any view you have of infinity is a finite algorithm generating a countable subset of infinity. That is not the same thing as an actual infinite object. – None – 2014-10-27T19:43:24.887

Also, I added the notion of the fractal function to avoid topology. Continuity means nothing to a Julia set unless the continuity extends to the dimension of the set itself. "Locally" is relative, and I can get behind any notion of "local" by just moving to a different notion of infinite complexity. – None – 2014-10-27T19:58:02.210