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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.

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  • Seems like no. There are an infinite amount of numbers between 0 and 1 without any repetition. – James Kingsbery Oct 27 '14 at 16:17
  • 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 Oct 27 '14 at 18:07
  • 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 Oct 27 '14 at 18:10
  • 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 Oct 27 '14 at 18:13
  • @MoziburUllah Each real number, and each natural number 1, 2, 3, ... is a distinct set when 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 Oct 27 '14 at 18:57
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I'm not sure the question is sufficiently precise. Let's ask: in an infinite sequence, must any given section repeat (i.e. have another section isomorphic to it)?

The answer here seems to me clearly to be no. Think of Cantor's diagonalization argument. I make up an infinite number of strings of 1s and 0s, each of which is infinitely long like so:

S1 1, 1, 1, 1, 1, . . .

S2 1, 0, 1, 0, 1, . . .

S3 0, 1, 0, 1, 0, . . .

S4 1, 0, 0, 1, 0, . . .

S5 0, 1, 1, 0, 1, . . .

Now I diagonalize to create a new sequence Sx, whose first value is opposite of the first value of S1, whose second value is the opposite of thesecond value of S2 and so on.

Sx 0, 1, 1, 0, 0, . . .

Now I know that Sx has not occurred anywhere in any of the infinite number of sequences S1 . . . Sn. (Via proof by contradiction. Suppose Sx is identical to some sequence Sm, then it has to be the case that the mth value of Sx is the opposite of the mth value of Sm, because this is how Sx was defined. Contradiction.) But if I know that Sx isn't among any of the S1 . . . Sn, then I also know this sequence hasn't been repeated anywhere in that infinite set of sequences.

I feel like there should be an application of this fact to our question about whether a section of an infinite sequence should have to repeat or not. Perhaps somebody else will see how to connect this last link for us.

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  • This is a nice generalisation of the diagonal method that I haven't seen before; however you've only shown that Sx isn't identical to any of the S1, S2,...; but it may be the case that Sx is Some subsequence of S1 or a translation; but your argument ignores a crucial distinction that Spinoza insists on that is essence; is one sequence of digits essentially different from another? – Mozibur Ullah Oct 27 '14 at 20:59
  • To put it in a different form are all the performances of the play The Merchant of Venice essentially different from each other; or essentially the same; Spinozas answer might suffer here from a theatre critics who is looking for different things. – Mozibur Ullah Oct 27 '14 at 21:02
  • @MoziburUllah All the 'subsequences and translations' of Sn are in the list, too, and each of them differs from Sx in some position. This is the standard proof the Reals are not countable. Any way in which these are essentially the same involves abstracting away information. Classical math does that only on purpose by creating equivalence classes. So unless I have decided to do so, these are all essentially different. By construction, the 'essence' of a real number is formally equivalent to its binary expansion. – user9166 Oct 27 '14 at 21:38
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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?

Yes to the latter. To understand the infinite, I find it best to first get a firm grip on the notion of the finite.

Consider the following non-numerical analogy: Suppose we start with a walk through an ordinary (finite) village comprised of several houses. Suppose further that you are free to walk around this village to visit some or all those houses in any order, as you choose.

If you start at one house, and keep going from one house to another, and go to no house more than once, it stands to reason that you must eventually return to your starting point. You must eventually run out of different places to go. Intuitively, this would be true of any finite village. This would not be true in an "infinite" village where you could start at one house and never return to it on your walk, even if you could walk for an eternity. Not surprisingly then, a village (or other set of objects) is infinite if and only it is NOT finite.

A set of objects can be said to be infinite if and only if it is possible to start at some element and keep going from one element to another, and to not go to any element more than once (i.e. no repetition), and never return to the starting point.

(Also see "Infinity: The Story So Far", revised just now at my math blog, for a formal development of these ideas.)

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Please see my response to this similar question here.

Summary: If the set of possible states is bounded, then some state must recur. But if not, not. The set of states 1, 2, 3, 4, ... never repeats. But if you say that there are only 3 states, then some state must repeat infinitely many times. But it is not necessarily the case that all states will recur. So for example if there are infinitely many universes and only finitely many legal states available in a region of spacetime, then somebody has infinitely many copies of themselves out there; but most likely not me, and most likely not you. Like the kids say these days: YOLO. You only live once. They don't realize this, but they are making a deep and insightful point about cosmology and metaphysics.

But please do see my original response, which also considers the probabilistic argument often seen online. Long story short, probability zero events can occur in infinite probability spaces; therefore the probabilistic argument that we all live many times is false.

And besides: We don't know what constraints are imposed by the laws of physics on the set of allowable states. So the naive argument that "Everything must happen in an infinite sample space" is just false.

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I would claim that infinity and repetition have nothing to do with one another at a basic level, but that every human representation of infinity can only be made via repetition.

We have only finitely many symbols to work with, and via them we can define only countably many things with clarity. We can define the integers, and give each of them a unique representation, and go from there to the rationals, and from there to all algebraic roots of rational polynomials, and from there to all closed integral forms with bounds among those algebraically identified numbers, etc. etc. etc. After a countably infinite number of layers, we could eventually have a representation for every real number, in theory.

But if we really want to write down an allusion to infinity, it has to be in terms of a finite set from among this tower of more-and-more complex representations, and beyond that, if it is really going to be deterministic, it has to be captured in some finite algorithm that would write out the rest of the representations to which we are alluding. Therefore loops or recursion, and therefore repetition.

When we think that the points of the real line all 'look the same', it is because the vast majority of them cannot have names that would focus our attention on them in a way that would make a difference. But this is an illusion forced on us by language. We cannot capture the detail in any meaningful way with finitely many symbols.

(From a constructivist point of view, that means most of those points do not exist, and everything is finite with a single countable iteration represented as a process. Infinite constructs are useful for projecting concepts onto, but if you cannot construct the results, you have no object. From that point of view, all infinities are countable, and inaccessible. So in that frame of mind (which I manage at my best), your observation is correct.

Still, being correct in reality and being correct in principle are not the same, this association is false in principle.)

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  • Comments are not for extended discussion; this conversation has been moved to chat. – stoicfury Oct 29 '14 at 1:53
  • I truly regret my participation in this afternoon's conversation. I'll be taking a break from this forum for a while. As it happens, I suspected that given the topology on the reals you can recover the order; and this does in fact turn out to be true. In other words: the topology on the reals does NOT "forget" the order. See y'all around. math.stackexchange.com/questions/995984/… – user4894 Oct 29 '14 at 2:15
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What is a repetition?

Popular examples so far include numbers and physical objects being counted, but it is all fraught with space-time conceptions. A fraction of a number carrying on infinitely is not infinity. Supposing that 1 unit may be added or subtracted infinitely is not infinity.

When defining infinity, why believe that we can apply space-time principles at all?

Infinity does not exist in space-time, so we should not imply time when describing it; therefore, no, infinity must not involve repetitions.

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  • If you have references to people who take a similar view this would support your answer and give the reader a place to go for more information. Welcome! – Frank Hubeny May 10 at 21:27
  • almost an interesting answer tho IMHO @FrankHubeny ! – another_name May 10 at 21:37
  • I have not yet sought supporting viewpoints, though I do assume they exist. – Joseph Ryle May 11 at 0:05

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