# Assuming an Infinite amount of time, are all things inevitable? **URL:** \ **Category:** Factual Questions\ **Created:** [January 8, 2012, 9:33pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850 "2012-01-08T21:33:00Z")\ **Posts on this page:** 20\ **Page:** 4
**Author:** ![WarmNPrickly](https://avatars.discourse-cdn.com/v4/letter/w/d78d45/32.png) [@WarmNPrickly](https://boards.straightdope.com/u/WarmNPrickly)\ **Post date:** [January 11, 2012, 4:01pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/61 "2012-01-11T16:01:41Z")
> [@Exapno\_Mapcase](#): > > This is true only if space is not quantized. That’s why I specified Planck length and Planck time above. Once you have a minimum physical distance and time, then only a finite number of states are possible, as I don’t know enough about the Planck length to know if it even applies in this instance. I’m not certain that the Planck length would imply that all distances are inherently quantized. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [January 11, 2012, 4:06pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/62 "2012-01-11T16:06:07Z")
> [@WarmNPrickly](#): > > I don’t know enough about the Planck length to know if it even applies in this instance. I’m not certain that the Planck length would imply that all distances are inherently quantized. It’s an assumption. If the universe is quantized then it has a minimum length and minimum time (since spacetime is a single thing). We don’t know if that minimum is equal to the Planck length and time because current theory doesn’t take it that far. However, the principle for the calculations is identical. Whatever the minimum is can be plugged into the equation to give the total (finite) volume and the total (finite) units of time and those equal the total (finite) number of states. ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [January 11, 2012, 4:28pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/63 "2012-01-11T16:28:26Z")
> [@Exapno\_Mapcase](#): > > This could happen, in the sense that 1/7 has an infinite but repeating decimal expansion. My point, again, is that a random assortment of universes in infinity can be assumed to be like the relationship between the rationals and reals. There are so many more reals between 0 and 1 than rationals that the likelihood of picking a rational randomly is 0. Once you start playing around with an infinity of universes you have to accept that all possible states will occur each an infinite number of times. You can’t get out of it. Your proof isn’t a proof. The question is whether all things are inevitable in an infinite amount of time. I demonstrated an example of an infinite amount of time which did not contain all things. It’s a proof by contradiction. I don’t have to prove my possibility occurs for this purpose; it’s sufficient to show it’s possible. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [January 11, 2012, 4:30pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/64 "2012-01-11T16:30:31Z")
If the universe isn’t really quantized, but we are absolutely constrained to measure it in a quantized fashion by the limitation of what’s possible at the bottom end of the scale, the effect is the same. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [January 11, 2012, 4:37pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/65 "2012-01-11T16:37:17Z")
> [@Little\_Nemo](#): > > And in this infinity, there will never be a performance of Shakespeare. So this proves that an infinite amount of time does not guarantee everything will eventually happen. I think there will - because it’s entirely dependent on your encoding paradigm, which is artificial and imposed anyway. 0110 is six in binary, but you could just as easily define an encoding paradigm that divides up different-length strings of zeroes and assigns meaning to them - it doesn’t seem any more or less valid than picking out arbitrary meaning from binary interpretation of a collection of random ones and zeroes. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [January 11, 2012, 4:49pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/66 "2012-01-11T16:49:09Z")
> [@Little\_Nemo](#): > > The question is whether all things are inevitable in an infinite amount of time. I demonstrated an example of an infinite amount of time which did not contain all things. It’s a proof by contradiction. I don’t have to prove my possibility occurs for this purpose; it’s sufficient to show it’s possible. This isn’t a math proof we’re dealing with, though. It’s a probability proof. Showing one counter-example isn’t sufficient. What I’m saying is that there will indeed be not just one but an infinity of counter-examples that are repetitive (there will also be an infinity of examples that are essentially equal to 0) but since there will also be an infinity of examples which are \*\*not \*\*repetitive then all possible things \*\*must \*\*occur infinitely often. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 11, 2012, 6:27pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/67 "2012-01-11T18:27:31Z")
**Exapno** , you seem to be using the language of probability and possibility such that “has probability zero\*” and “is impossible” are identified for you. **Little Nemo** is not using the language of probability and possibility this way. Indeed, it is mathematically non-standard to use the language of probability and possibility this way, although you are free to, if you like. (To use the language in this way has some subtle difficulties which take some technical cleverness to reconcile, however; for example, suppose you pick a random number from [0, 1]. Whatever number you pick, that particular number had a prior probability of 0 of being picked. Yet you picked it all the same! So how could that event of probability 0 have been impossible?) On the standard way of speaking, events of probability 0 are not impossible; they are just very improbable (to the point of “almost certainly” not happening, but not to the point of certainly not happening). [\*: (on the standard account whereby all infinitesimal probabilities are zero)] ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 11, 2012, 6:33pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/68 "2012-01-11T18:33:04Z")
We had a thread a while back that brought up a lot of these issues, like the difference between “probability 0” and “impossible,” the different sizes of infinity, and the ever-popular infinite monkeys with typewriters: [“Anything that can happen, will happen, given enough time.” – Really?](http://boards.straightdope.com/sdmb/showthread.php?t=520614) ---
**Author:** ![TimeWinder](https://avatars.discourse-cdn.com/v4/letter/t/bcef8e/32.png) [@TimeWinder](https://boards.straightdope.com/u/TimeWinder)\ **Post date:** [January 11, 2012, 6:50pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/69 "2012-01-11T18:50:43Z")
> [@Mangetout](#): > > If an event is of sufficient rarity that it can only occur a finite number of times in an infinite period, that means the infinite period can be divided into two separate infinite periods, in one of which, the event doesn’t occur at all. You’re confusing “very large” and “infinite” again. There’s no such thing as an event of sufficient rarity that it can only occur a finite number of times in an infinite period. If the event occurs with non-zero probability, it will occur an infinite number of times. If it occurs with zero probability, then it won’t occur at all. (This does assume that the probability of the event can’t change over time). The question boils down to: can you determine if the probability of the event is non-zero? That’s the real question with Pi, for example. It’s possible for a digit sequence to be infinite and non-repeating ( 0.12112211122211112222…) and yet have a valid sequence (“3”) have a zero probability of occurring. Fundamentally, the answer to the OP in our universe is “no” – even if the universe is infinite, the matter in it isn’t – it’s eventually subject to proton decay, and thus the “infinite period” for events involving matter is nicely finite. To pervert Douglas Adam’s famous quote, “Space is _merely_ mind-bogglingly large.” ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 11, 2012, 6:55pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/70 "2012-01-11T18:55:34Z")
> [@TimeWinder](#): > > There’s no such thing as an event of sufficient rarity that it can only occur a finite number of times in an infinite period. Why not? Most things really only occur once, and wouldn’t reoccur no matter how much time you give them. > [@](#): > > If it occurs with zero probability, then it won’t occur at all. This is incorrect: see **Indistinguishable** ’s post above. > [@](#): > > The question boils down to: can you determine if the probability of the event is non-zero? That’s the real question with Pi, for example. It’s possible for a digit sequence to be infinite and non-repeating ( 0.12112211122211112222…) and yet have a valid sequence (“3”) have a zero probability of occurring. It’s also possible for a digit sequence to be infinite and non-repeating and have “3” occur exactly once, (0.312112211122211112222…), or twice, or any particular finite number of times. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [January 11, 2012, 7:40pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/71 "2012-01-11T19:40:17Z")
> [@TimeWinder](#): > > You’re confusing “very large” and “infinite” again. I don’t think I am, because the next thing you said is exactly the point I was trying to make: > [@](#): > > There’s no such thing as an event of sufficient rarity that it can only occur a finite number of times in an infinite period. ---
**Author:** ![TimeWinder](https://avatars.discourse-cdn.com/v4/letter/t/bcef8e/32.png) [@TimeWinder](https://boards.straightdope.com/u/TimeWinder)\ **Post date:** [January 11, 2012, 7:46pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/72 "2012-01-11T19:46:21Z")
> [@Thudlow\_Boink](#): > > Why not? Most things really only occur once, and wouldn’t reoccur no matter how much time you give them. Then the probability has been changed by the original event, as I noted. If the event still has non-zero probability after happening, then it _will_ reoccur, in fact, an infinite number of times. Events whose occurrence alters their probability are interesting, and perhaps common, but since a single incident will answer the OP’s question, not very relevant. More relevant are events whose probability changes over time, and even these only matter if that probability eventually goes to “impossible” – as in the case of interactions of normal matter in a universe where all normal matter will eventually be gone. > [@](#): > > This is incorrect: see **Indistinguishable** ’s post above. Indistinguishable is talking about events which have probability zero as a limit. To get around that quibble, replace “have probability zero” with “are impossible”. > [@](#): > > It’s also possible for a digit sequence to be infinite and non-repeating and have “3” occur exactly once, (0.312112211122211112222…), or twice, or any particular finite number of times. Sure, but that’s not the point I was making. Events are either possible or they aren’t. If they are, they must happen in an infinite space, but not a finite one, no matter how large. The trick isn’t determining whether a rare event will occur, but rather whether or not it’s impossible or merely improbable. ---
**Author:** ![TimeWinder](https://avatars.discourse-cdn.com/v4/letter/t/bcef8e/32.png) [@TimeWinder](https://boards.straightdope.com/u/TimeWinder)\ **Post date:** [January 11, 2012, 7:48pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/73 "2012-01-11T19:48:56Z")
> [@Mangetout](#): > > I don’t think I am, because the next thing you said is exactly the point I was trying to make: Yeah, sorry. The post to which I was responding to could be read as arguing for either position, and I picked the wrong one. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 11, 2012, 7:52pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/74 "2012-01-11T19:52:12Z")
> [@TimeWinder](#): > > Indistinguishable is talking about events which have probability zero as a limit. Why do you say that? In standard mathematical terminology, “has probability zero” and “is impossible” are not the same (the latter implies the former, but not vice versa). “has positive probability” and “is possible” are not the same (the former implies the latter, but not vice versa). “has probability zero” and “will never occur, over infinitely many independent trials” are not the same. “has positive probability” and “will definitely occur, over infinitely many independent trials” are not the same. In standard mathematical terminology, there is NO relationship between probability and possibility, except for that impossible events must have probability zero (but the converse of this is not assumed). You could have a coin with 100% heads probability, 0% tails probability, and have it flip to tails. I’m not saying you have to speak the standard way, mind you. But we should at least be aware of what we mean, and if we’re all using the same words in different ways, we need to understand that. Math can’t prove anything that isn’t true by definition alone. For some in this thread, it’s hardly definitional that an infinite sequence of occurrences MUST contain every possible occurrence infinitely often (hence all the invocation of sequences like “0.311122121122111221212”). For others, they are using “MUST” and so on in a different way. The whole argument would evaporate if people realized this. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 11, 2012, 7:59pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/75 "2012-01-11T19:59:38Z")
> [@Indistinguishable](#): > > For some in this thread, it’s hardly definitional that an infinite sequence of occurrences MUST contain every possible occurrence infinitely often (hence all the invocation of sequences like “0.311122121122111221212\*\*…\*\*”). Missing ellipsis reinstated in bold. > [@Indistinguishable](#): > > But we need to at least understand what we mean when we speak, however we speak, and if we’re all using the same words in different ways, we should be aware of that. Quote altered to what I wanted to say but lost the edit window for. ---
**Author:** ![TimeWinder](https://avatars.discourse-cdn.com/v4/letter/t/bcef8e/32.png) [@TimeWinder](https://boards.straightdope.com/u/TimeWinder)\ **Post date:** [January 11, 2012, 8:05pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/76 "2012-01-11T20:05:15Z")
I think several of us are trying to solve two problems at once: the difference between the real world and a mathematical abstraction of one, and a language issue at the borders. Specifically, the claim about random numbers in an interval having probability zero: If my random numbers are required to be, say, integers, and my range is from 1 to 10, then the probability of my picking a given number is 1/10. If the range is from 1 to 100, then it’s 1/100. It’s only in the limit, when the range becomes infinite, that the probability actually becomes zero. That’s nice and all, but a significant distinction from picking, say, 3.5, which has probability “impossible” all along – it’s just not in the set of valid outcomes. At the other end of the spectrum, we have the converse. The only reason “possible” doesn’t imply “non-zero possibility” is that the possibility may have gone to zero as the limit of an infinite number of options. So can we get along with “is able to occur” and “is not able to occur” or something, and start from there? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 11, 2012, 8:06pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/77 "2012-01-11T20:06:39Z")
> [@TimeWinder](#): > > Sure, but that’s not the point I was making. Events are either possible or they aren’t. If they are, they must happen in an infinite space, but not a finite one, no matter how large. The trick isn’t determining whether a rare event will occur, but rather whether or not it’s impossible or merely improbable. This suggests your argument that “[Every possible event] must happen in an infinite space, but not a finite one, no matter how large” has nothing to do with probability, and is purely to do with the concept of possibility, in itself. Ok. Fine. But this is a shocking statement to me, in a non-probabilistic context. What is the argument? Why must every possible event happen in an infinite space (but not a finite one, no matter how large)? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 11, 2012, 8:31pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/78 "2012-01-11T20:31:58Z")
> [@TimeWinder](#): > > At the other end of the spectrum, we have the converse. The only reason “possible” doesn’t imply “non-zero possibility” is that the possibility may have gone to zero as the limit of an infinite number of options. I wouldn’t say that’s the only reason. The reason is just that probability and possibility are not assumed to have anything at all to do with each other, anymore than probability has anything to do with cleanliness or godliness or tigers or spelling. For that matter, these both have almost nothing to do with what actually occurs. The only relationships (in a standard setup where one happens to discuss all three at once) are that impossible things do not actually occur, and impossible things have probability 0. The converses of these are not assumed, and there is no relationship at all between “probability” and “what actually happens”. As I said, standard mathematical terminology allows straight-up for a two-sided coin, heads with probability 100%, and the other side with probability 0%, flipped just once, and coming up tails. There are no limits here, no infinite numbers of options. You may say “Well, fine, but the only reason probabilists did not adopt the convention of standardly identifying ‘possible’ and ‘has positive probability’ is to accommodate infinite limit cases”; that might be more on the mark. I will extend you that sympathy. [But, mind you, adopting such a convention is very, very difficult to make sense of, and would most likely lead one to want positive but infinitesimal probabilities, obviating the argument that an infinite product of probabilities less than 1 must always go to 0.] But what of it? Regardless of how they use their particular language, and you use yours, what is the actual argument that “Anything which is possible MUST occur infinitely often over infinite time”? Suppose I pick a countably infinite sequence of independent random numbers from [0, 1]. For each number in there, it is possible, each time, that I pick it. Yet in the end, no matter what, I’ve only picked countably many numbers; there are huge swathes of possible numbers that I never picked. How do you reconcile that with the claim that “Anything which is possible MUST occur over infinite time?”. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [January 11, 2012, 10:32pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/79 "2012-01-11T22:32:52Z")
> [@Indistinguishable](#): > > But what of it? Regardless of how they use their particular language, and you use yours, what is the actual argument that “Anything which is possible MUST occur infinitely often over infinite time”? > > Suppose I pick a countably infinite sequence of independent random numbers from [0, 1]. For each number in there, it is possible, each time, that I pick it. Yet in the end, no matter what, I’ve only picked countably many numbers; there are huge swathes of possible numbers that I never picked. > > How do you reconcile that with the claim that “Anything which is possible MUST occur over infinite time?”. The claim being made is that any finite string of numbers must be found and must be found infinitely often in a normal infinite expression. (Expression may not be the technical term.) A finite universe can be represented with a finite number string. Therefore any finite universe must be found and must be found infinitely often in infinite time and space. You seem to be doing something different, trying to pick a particular finite universe from the whole. That has probability zero, true. But it’s not the claim. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [January 11, 2012, 10:42pm UTC](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850/80 "2012-01-11T22:42:47Z")
> [@Exapno\_Mapcase](#): > > The claim being made is that any finite string of numbers must be found and must be found infinitely often in a normal infinite expression. Sure, by the definition of “normality”. By why must the sequence of occurrences over infinite time (or space or what have you) be “normal” (in this particular technical sense)? Why can’t it be non-“normal”? [Previous page](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850.md?page=3) [Next page](https://boards.straightdope.com/t/assuming-an-infinite-amount-of-time-are-all-things-inevitable/608850.md?page=5)