Caveat 1: I wrote this short essay for a forum of non-science types in the hopes of making them think different, and was hoping I might get some feedback in a respectable forum. Would love your thoughts...
Caveat 2: I am neither a mathematician or a physicist (software engineer), so some terms may only be partially defined, or is left to the reader to research more, so please take my banter with a grain of salt....

If you really think about it, the results can be mind bending. Consider that infinity does in fact exists, and is not just "a very large number". Next, consider a law of thermodynamics, Conservation of Mass (http://en.wikipedia.org/wiki/Conservation_of_mass), specifically. In a nutshell, this law states that all matter/mass remains constant over time...in layman's terms, matter can neither be created or destroyed. Next, realize that all matter is comprised of atoms (and other even smaller particles). If this is true, and it is a law we are discussing, the matter that comprises are bodies cannot be destroyed, it can be merely 'rearranged', into other forms of matter in the universe.

Extending this thought, our bodies are just one large (very large) collection of atoms. If infinity does exist, it is only a matter of time, albeit a very, very long time, math proves that these atoms will inevitably find themselves in the same arrangement they are in now. To help imagine this, think of a deck of cards. With three cards, we can create six unique combinations before we begin to repeat a pattern. Given four cards, we can create 24 unique combinations. Take the whole deck, and you get 52! (8.06581752E+67) different unique combinations. Now, it would take a really, really long time to create the same exact combination, but in it's essence, it is just a really big number. Two decks? 104! and repeat.

Just those two high-level concepts are enough to illustrate the conundrum. If infinity is real, and not just a really, really, large finite number, you almost don't even need to include the complexity of multiverses vs. a single universe....it just sometimes helps the medicine go down a little easier. One other thing, if infinity is real, it does not only exist into the future, it must also have existed throughout the past.

Now, if we pull all that together, you may begin to realize that.....everything you have done throughout your life, every decision made, and unmade, and every permutation of each, has both already been done, and will also happen again, over and over.

A rather athropocentric view of infinity. If the cosmos has any purpose or function I'm sure of one thing: It has fuck all to do with us any more than it has to do with the average beetle.

If you want to imagineer your way around the cosmos there's one easy way of making your considerations more realistic - imagine we and the planet Earth never existed.

Short answer: No.

Long answer: That´s what happens if you don´t have your infinities straight. Or make additional assumptions that aren´t made clear. The issues are as follows:
Arrangements of a finite number n of particles in 3d space during an interval T can be represented as a vector in R3nx[0,1], this set is infinite. Moreover it is uncountably infinite. There´s one such set for a particular decision you´ve made for instance. Now, given a universe infinite in duration, there have been countably infinitely many trials. Unless you assume that there is a probability greater than 0 for some particular states, this means the probability a particular vector has reoccured is 0. Even if there were countably infinite universes, the probability of reoccurence in one of them is still 0. If you assume there are states with strictly positive probabilities, they will repeat infinitely often, but there can only be countably many of them.
And that´s just the issues with the maths part.

This is exactly the kind of feedback I was hoping to garner. Makes me want to go back and get my math or physics degree. I should have paid more attention.

Thanks for the feedback so far all!

Birch

hoopy frood wrote:A rather athropocentric view of infinity. If the cosmos has any purpose or function I'm sure of one thing: It has fuck all to do with us any more than it has to do with the average beetle.

If you want to imagineer your way around the cosmos there's one easy way of making your considerations more realistic - imagine we and the planet Earth never existed.

Agreed, a very anthropocentric view...but I like having existed so far. I do like your proposition though, to envision we or the planet never existed. In my original thinking that would have to be one of the permutations that would have had to have existed I think.

Weaver wrote:I think you are mis-applying the conservation of matter/mass law of thermodynamics - it only applies to isolated systems.

We do not live in an isolated system.

This is where my noticeable lack of knowledge of details really shines. I noticed some verbiage stating that this may be thought of differently in special vs general relativity. Any merit to that? I will continue to read. My initial gut, which in this case does not say much, would be that in a single universe scenario, it could be defined as a closed system...inflationary yes, but still 'seems' closed to me.

Ok, where is my physics 101 book?

Birch

birchsport wrote:This is exactly the kind of feedback I was hoping to garner.

Are you familiar with Poincare's recurrence theorem?

ughaibu wrote:

birchsport wrote:This is exactly the kind of feedback I was hoping to garner.

Are you familiar with Poincare's recurrence theorem?

I was pointed towards that theorem and it sounded very similar. It is still on my list to finish reading as well as the references.

Birch

birchsport wrote:

Weaver wrote:I think you are mis-applying the conservation of matter/mass law of thermodynamics - it only applies to isolated systems.

We do not live in an isolated system.

This is where my noticeable lack of knowledge of details really shines. I noticed some verbiage stating that this may be thought of differently in special vs general relativity. Any merit to that? I will continue to read. My initial gut, which in this case does not say much, would be that in a single universe scenario, it could be defined as a closed system...inflationary yes, but still 'seems' closed to me.

Ok, where is my physics 101 book?

Birch

You're confusing a closed thermodynamic system with an isolated system. There is a reasonably detailed exposition of this here.

I would disagree with Weaver above, and suggest that a) conservation of mass/energy applies universally and b) there is probably no such thing as an isolated system, except the universe itself (by which I don't mean 'that which arose from the big bang'; more on that in the linked essay, as well as much of my writing on the topic of 'universe', which can be found using the forum search facility fairly easily).

Edited to add: There is a really excellent debunking of Kalamity Kraig's tired 'real infinities' argument available for download here (pdf). Highly recommended if you wish to understand infinities in philosophy and mathematics.

Conservation of Mass/Energy simply states that total mass energy is a constant, not the mass/energy cannot be created or destroyed. Note that Gravitational Potential Energy (GPE) is Negative, It is thought by many that including Gravitational Potential Energy, the total Mass/Energy in the universe is zero. As the universe expands, GPE and conventional Mass/Energy both increase canceling each other out.

I see no reason to rule out the existence of infinite sets in nature. In fact I could make a complexity argument that nature must be infinite since only infinite (or null) sets can be simple.

susu.exp wrote:Short answer: No.

Long answer: That´s what happens if you don´t have your infinities straight. Or make additional assumptions that aren´t made clear. The issues are as follows:
Arrangements of a finite number n of particles in 3d space during an interval T can be represented as a vector in R3nx[0,1], this set is infinite. Moreover it is uncountably infinite. There´s one such set for a particular decision you´ve made for instance. Now, given a universe infinite in duration, there have been countably infinitely many trials. Unless you assume that there is a probability greater than 0 for some particular states, this means the probability a particular vector has reoccured is 0. Even if there were countably infinite universes, the probability of reoccurence in one of them is still 0. If you assume there are states with strictly positive probabilities, they will repeat infinitely often, but there can only be countably many of them.
And that´s just the issues with the maths part.

I was having almost exactly this conversation* with a colleague only a couple of days ago at a Christmas lunch!
I put a view fairly similar to the OP, and then (because I sometimes can't help arguing with myself) posed the objection you made. My colleague's reply was that, while my objection was correct that the probability of any given configuration C0 recurring exactly is zero, for any given epsilon>0, however small, there would be infinitely many instances of configurations C(k) (k = 1 to infinity) for which the greatest distance between the position of any particle in C0 and its position in C(k) is less than epsilon. So there would be infinitely many instances of configurations that were indistinguishable from C0.

I though that was a pretty impressive objection given the number of beers he consumed while I was talking to him - and goodness know how many beforehand.

* It wasn't exactly the same conversation. My scenario was one considering the consequences if the cosmological constant's value dictates, via the FLRW cosmological equation, an infinite universe, and suggesting that that might mean there were an infinite number of people indistinguishable to us two, having exactly the same conversation at an indistinguishable Christmas party.

birchsport wrote:if infinity is real, it does not only exist into the future, it must also have existed throughout the past.

This doesn't follow. It is possible that time in our universe extends infinitely into the future but only finitely into the past. This could be the case either with a spatially finite or infinite universe. What is the case for the past is currently unknown (everything 'before' the Planck time is just speculation), despite eminent cosmologist William Lane Craig's best attempts to argue to the contrary.

andrewk wrote:I was having almost exactly this conversation* with a colleague only a couple of days ago at a Christmas lunch!
I put a view fairly similar to the OP, and then (because I sometimes can't help arguing with myself) posed the objection you made. My colleague's reply was that, while my objection was correct that the probability of any given configuration C0 recurring exactly is zero, for any given epsilon>0, however small, there would be infinitely many instances of configurations C(k) (k = 1 to infinity) for which the greatest distance between the position of any particle in C0 and its position in C(k) is less than epsilon. So there would be infinitely many instances of configurations that were indistinguishable from C0.

Well, if you discretize the state space, you of course end up with a countably infinite number of states. But in this case you basically don´t check whether C(t)=C(0) but whether C(t)~C(0) with ~ an equivalence relation. In this case it´s worth noting that you then get a discrete stochastic process in discrete time and while you can reasonably assume that it´s markov, this means that states are either transient or reccurent. If it´s furthermore irreducible then either all states are reccurent or all states are transient, if it is not irreducible then there can be one or more recurrence classes - i.e. sets of states that repeat, but only if one member is reached (an example for a reducible process is a general model of evolution. Here all states are transient and do not repeat, with the exception of the state where the population in question has a size of 0). There´s a whole lot of argument that needs to be made before you can state that there´s a reccurence for the discrete equivalence classes.