Infinite amount of monkeys playing chess

I will prove the statement (the first sentence) false:

"All possibilities" can only be accessed through the Set of Everything.

The Set of Everything does not exist. QED

If you meant "All possible chess games," this argument doesn't hold (that is, the truth of the statement goes undetermined until further argument).

I will deliver a proof that the Set of Everything doesn't exist upon request.

You are the reincarnation of Bertrand Russell, AICMFP.

I have a pet monkey named Ivan who was trained by a Russian Grandmaster.  I paid too much because I consistantly beat him using the Scandanavian.  At best, the monkey can only win once in three games. 

It's worth remarking that many facts in physics are not about single examples but about the statistics of large numbers of similar examples.

Two key examples are the uncertainty principle and the second law of thermodynamics. The first is really about a large number of parallel experiments in which position and momentum are measured. For a single experiment all you can do is predict the probabilities of different results. The second is about a large number of examples of the same system which evolves from an initial state to a final state.

The point of this is that in a single system (say a single Universe) there is no law that says that a high entropy state will be reached. It is just that it is very much more likely than a low entropy state.

To simplify, suppose there is a idealised isolated system, a room with an ideal gas in it, which all starts at one side of the room (a rather low entropy state). We all know at some time in the distant future the gas is likely to be evenly distributed at a single temperature and pressure. But leave the gas there for an infinite time it is certain that at some time the gas will happen to all temporarily end up on the same side. This is not to say that everything you can imagine will happen. Energy is conserved, which means that when the gas all ends up on the same side of the room, its temperature must be higher.

"Entropy increases" could be expressed as "systems are more likely to move towards more general states than special states". Eg there are a much larger number of states of the gas in a room where it is distributed evenly than the number where it is all on one side of the room. To be precise, about {2 to the power of the number of molecules} times as many states. The more common states are more likely to occur merely because there are more of them.

The conclusion of this was meant to be that entropy increasing does not really stop monkeys from beating Carlsen, but to be frank, this conclusion is of far less interest than the reasoning.

So what would you consider to be the ideal gas?

I'm aware of a number of gasses that are noble....

A thing will only happen in an infinite time if there is a non-zero chance of it occuring. A monkey can't fly, because if it has the appendages to fly, it won't be a monkey. The better analogy with the uncertaint principle is dxdp, where x and p are position and momentum. I.E. If we know the momentum exactly, we cannot find the object. But even that doesn't count as "flying".

My understanding is that all transitions that obey the conservation laws of quantum mechanics will occur eventually. So a monkey will at some time turn into anything that could be made out of it without breaking the laws of conservation of energy, hadron number, charge, etc. For example, it will definitely turn into a graphene-based computer (plus some left over stuff). This is of course highly improbable, but certain if you wait long enough.

[I make the usual assumption about a closed system]

So know we have gotten to monkeys magically transforming into chess playing computers over time?

I think that they may have lost me somewhere along the way :/

Not unlike this:

http://en.wikipedia.org/wiki/Boltzmann_brain

It may well be that in addition to a Boltzmann Brain or two, there have also spontaneously appeared a handful of perfect chess playing monkeys.

I'll fill in a few of the details in my claim.

The laws of particle physics can be roughly summarised as there being a chance of anything happening that is not prohibited by conservation laws. At the lowest level this comes down to particles interacting/transforming into other particles.

Well, these laws apply for all objects since they are made up of particles. So if the conservation laws (conservation of energy, charge and so on) don't prohibit a transformation, there is a probability of it happening. Of course things like monkeys transforming into graphene computers have an absolutely absurdly low probability of happening. But if you are going to imagine a closed system with a monkey in it for an infinite time, such things are going to happen with probability 1, just like all the molecules of gas in a room finding themselves on the same side. It's like entering the lottery a trillion times - you are almost certain to win eventually.

Can an infinite number of monkeys even exist in a finite universe?

Are there an infinite number of better tasks for an infinite number of monkeys to accomplish?

Clearly not if it is bounded both in space and time. But if it is unbounded in time, there's room for an infinite number of monkeys with finite lifetimes to exist.

But if only one at a time, they can't play each other.

You're correct that it's not possible to prove infinity, because everything that we, as humans, know about only has a finite lifetime. It's only a concept that people have invented.

Like throwing an infinite amount of feces at an infinite number of walls until they reproduce the complete works of Dan Brown.

Shouldn 't take longer than a week or two.....

I'll say again, look into the Cantor Set:

http://www.helium.com/items/788107-cantors-infinity-proof-made-easy