# Infinite monkey theorem - question(s)

by Thomas1989
Tags: infinite, monkey, theorem
 P: 21 Hello everyone, glad to be here. I'm Tom, 22 from Manchester UK. I don't consider myself intelligent, I failed miserably at GCSE Maths but I tend to think a lot about time and space, and I'm interested in the subject and keen to learn more. I'd like to hear some peoples opinions on the topic of time and the infinite monkey theorem. Here is my question(s). Recently me and my friend were talking about if we think time is infinite. In all honesty I don't have a clue, but for some reason I feel naturally inclined to say infinite. So, is there any sort of consensus in the scientific community? Or is it still widely debated? More importantly, my friend brought up the infinite monkey theorem. I had never heard of it before, and this really got me thinking. If time really is infinite, would it be possible over an inconceivable amount of time for something similar to the infinite monkey theorem to happen on a much bigger scale? For example, is it theoretically possible that everything that has ever happened so far, Earth, our existence, and all events could happen again in exactly the same sequence? If time really is infinite, of course... I can't get my head around the notion of eternity and an infinite space/time. I'm also a guy struggling to find my way between Science and Faith, and I have lately been swaying to Creationism, but that's a different matter entirely. Would appreciate any feedback, I'd like to know your thoughts on this! Cheers!
 P: 526 Hi Thomas, welcome to PF! The standard model of cosmology holds that dark energy (the cosmological constant) will continue to expand the universe, for seemingly forever. Now, this does not lead to an infinite monkey situation, because star formation will eventually end, and everything in the universe will eventually decay into particles. Here is an article to read. But, if the universe is spatially infinite, then everything that can happen, does happen somewhere. The issue of spatial size is one that is strongly debated, and will remain so for a while. There is no reason to go either way right now, there is no evidence for either position.
 P: 21 Cheers Mark! Nice to meet you. I understand! Must say, that's quite a morbid conclusion. So after the Big Freeze, what? Eternal darkness? Grim. Brilliant read though, thanks for the link. Shortly after I posted this, I found an example of what I was thinking of in The Eternal Return theory. It is racking my brain. Such a mind boggling subject, enjoying looking around the forums though. Cheers again!
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## Infinite monkey theorem - question(s)

In a cyclical universe model, after heat death the universe would somehow manage to undergo another big bang scenario at some unimaginably remote time in the future.
 P: 19 Hi, Thomas! Here's a talk you might like; Brian Greene is not the most scientific of scientists, but he addresses some of the concepts of an infinite universe that you're thinking of: http://www.radiolab.org/blogs/radiol...lti-universes/ Essentially, if space is infinite, then everything possible is going on in an infinite number of places at all times. This is only logical. But, even if time goes on forever from an initial starting point, as Mark M was getting at, the universe is moving in a definite direction, so the monkey situation will not occur. I've never even heard of the eternal return theory, but after looking it up, I will warn you to be skeptical of things like this. Ancient philosophical constructs are almost never in agreement with reality.
P: 1,638
 Quote by Thomas1989 If time really is infinite, would it be possible over an inconceivable amount of time for something similar to the infinite monkey theorem to happen on a much bigger scale? For example, is it theoretically possible that everything that has ever happened so far, Earth, our existence, and all events could happen again in exactly the same sequence?
You are not including jump-discontinuities in your logic. It's the "straw that breaks the camel's back". Things sometimes do not evolve smoothly but rather sometimes, at critical points, abruptly and qualitatively change state. When this happens, often the rules describing the system before the critical point are no longer applicapable to the system after the critical point: a bolt snaps on a bridge. It collapses. The speed limit over the bridge is no longer applicable to the pile of rubble. Critical-point change are all around us from the very small to the very large. Therefore I do not feel it's unreasonable to suggest the entire Universe and all that is in it also behaves that way including time: at some point in the evolution of the Universe, a critical point may be reached where time looses meaning. At that point time is neither finite nor infinite, it simply is no longer applicapable. A critical point was reached, a qualitative change occured and there simply is no time in the new state. So based on this argument, and all the evidence around me to suggest qualitative change occurs at critical points, I suggest time is not infinite.
P: 571
 Quote by Acala Hi, Thomas! Here's a talk you might like; Brian Greene is not the most scientific of scientists, but he addresses some of the concepts of an infinite universe that you're thinking of: http://www.radiolab.org/blogs/radiol...lti-universes/ Essentially, if space is infinite, then everything possible is going on in an infinite number of places at all times. This is only logical. But, even if time goes on forever from an initial starting point, as Mark M was getting at, the universe is moving in a definite direction, so the monkey situation will not occur. I've never even heard of the eternal return theory, but after looking it up, I will warn you to be skeptical of things like this. Ancient philosophical constructs are almost never in agreement with reality.

Greene is incorrect. I have discussed this argument before, but it is like trying to eradicate crabgrass.

The first thing you have to do is define what equality is. How do you know whether two things are the same or not? How close is good enough?

There seems to be an unstated assumption that the number of states of, say, a human being is finite. This I do not believe. Perhaps the assumption is that the number of states is countably infinite. That I do not believe either. All you need is one continuous quantity in the universe and it becomes uncountably infinite. All this needs to be cleared up before you can argue anything. I never seen anyone bother to lay this groundwork, possibly because no one knows the answer.

Another bogus argument I have seen is that if something exists then the probability it exists is greater than zero. In a finite Universe that is true, but in an infinite Universe it is not true. In an infinite Universe the probability is whatever it is. Observing the Earth tells you nothing about the probability of something the "same" as the Earth, because the Earth wasn't chosen at random.
 PF Patron Sci Advisor P: 8,880 The observable universe is not temporally infinite, therefore it is not spatially infinite. We cannot observe inaccessible regions of the universe so your argument is scientifically irrelevant.
Mentor
P: 9,588
 Quote by ImaLooser There seems to be an unstated assumption that the number of states of, say, a human being is finite. This I do not believe. Perhaps the assumption is that the number of states is countably infinite. That I do not believe either. All you need is one continuous quantity in the universe and it becomes uncountably infinite. All this needs to be cleared up before you can argue anything. I never seen anyone bother to lay this groundwork, possibly because no one knows the answer.
Quantum mechanics give you a natural measure of "similar" - if the difference between two things is too small, you cannot distinguish them like you could to this with macroscopic objects.
It does not matter which interpretation of QM you prefer, you can always divide your phase space in a countable number of regions with a volume of the minimal phase space a system needs (\hbar^whatever). This does not mean that these are different states (and your choice how to divide the phase space is arbitrary), but it does mean that you cannot get an uncountable amount of states which you could call "different" in a meaningful way.

Apart from that, I think that you could collect a lot of quantum states and still get the same "state of a human". It does not really matter if molecule Y is at position X or 1nm above that. For the state of mind, I think (nearly) identical connections between all cells in the brain and an identical activity pattern at a certain point in time would be sufficient.

@Thomas1989: While monkeys might die due to the lack of stars in the far future, you can always look at Boltzmann brains.
P: 19
 Quote by ImaLooser Greene is incorrect. I have discussed this argument before, but it is like trying to eradicate crabgrass. The first thing you have to do is define what equality is. How do you know whether two things are the same or not? How close is good enough?
I don't see the problem; identical is good enough.

 Quote by ImaLooser There seems to be an unstated assumption that the number of states of, say, a human being is finite. This I do not believe. Perhaps the assumption is that the number of states is countably infinite. That I do not believe either. All you need is one continuous quantity in the universe and it becomes uncountably infinite. All this needs to be cleared up before you can argue anything. I never seen anyone bother to lay this groundwork, possibly because no one knows the answer.
The assumptions you require are indeed necessary, and in this view we take the number of possible states as finite or countably infinite. I don't see how it can be demonstrated either way; I think it's a reasonable assumption to start with.

 Quote by ImaLooser Another bogus argument I have seen is that if something exists then the probability it exists is greater than zero. In a finite Universe that is true, but in an infinite Universe it is not true. In an infinite Universe the probability is whatever it is. Observing the Earth tells you nothing about the probability of something the "same" as the Earth, because the Earth wasn't chosen at random.
We aren't looking at probabilities of anything; we're just looking at whether it exists or not. I don't see what probability you're referring to. Even if the probability is zero, as you said, it can perfectly well still exist, and it does, within the bounds of our assumptions.
P: 571
 Quote by mfb Quantum mechanics give you a natural measure of "similar" - if the difference between two things is too small, you cannot distinguish them like you could to this with macroscopic objects. It does not matter which interpretation of QM you prefer, you can always divide your phase space in a countable number of regions with a volume of the minimal phase space a system needs (\hbar^whatever). This does not mean that these are different states (and your choice how to divide the phase space is arbitrary), but it does mean that you cannot get an uncountable amount of states which you could call "different" in a meaningful way. Apart from that, I think that you could collect a lot of quantum states and still get the same "state of a human". It does not really matter if molecule Y is at position X or 1nm above that. For the state of mind, I think (nearly) identical connections between all cells in the brain and an identical activity pattern at a certain point in time would be sufficient. @Thomas1989: While monkeys might die due to the lack of stars in the far future, you can always look at Boltzmann brains.
I can work with that. Let's forget about humans and do planets. Let's divide the phase space into a countably infinite number of disjoint regions. We don't know what the measure of any of these regions is.

The statement that everything happens an infinite number of times is equivalent to the statement that the set of events in each region is infinite. It is possible so it can't be disproved, but I don't see any particular reason to believe that. My position is that we know nothing about the sets in any of these regions. Some of them might even be empty.
Mentor
P: 9,588
 The statement that everything happens an infinite number of times is equivalent to the statement that the set of events in each region is infinite.
This is true in finite volumes of spacetime (and therefore phase space) only. And I think all the "everything happens an infinite number of times" assume infinite space or some infinite repetition in time.
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 Quote by Thomas1989 Cheers Mark! Nice to meet you. I understand! Must say, that's quite a morbid conclusion. So after the Big Freeze, what? Eternal darkness? Grim. Brilliant read though, thanks for the link.
Why? Is it any more morbid or grim that the universe will "come to a halt" after millions of years than the fact that you and everyone you know is going to die in a compartive eye-blink?

 Shortly after I posted this, I found an example of what I was thinking of in The Eternal Return theory. It is racking my brain. Such a mind boggling subject, enjoying looking around the forums though. Cheers again!
 P: 635 I'm surprised no one's brought the concept of GR implying that there must be some sort of "Big Bang" event into this.
P: 615
 Quote by Chronos The observable universe is not temporally infinite, therefore it is not spatially infinite. We cannot observe inaccessible regions of the universe so your argument is scientifically irrelevant.
The second part I understand, but the first part I am confused about.

When you say that it is not temporally infinite, is that because it started at some point? Are rays not considered to have infinite length?

And why does "it is not spatially infinite" follow from "it is not temporally infinite"?
 Mentor P: 9,588 The past is finite (as far as we know), as time started with the big bang 13.7 billion years ago. Now, imagine some point in space which emitted light right after the big bang. How far away from its initial position would it be by now? Due to the expansion of space, it turns out that this value is larger than 13.7 billion light years, something close to ~45 billion ly. This is the radius of the observable universe - the part of the universe we can observe today.

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