The proof is very simple really. Assume p(n) is the largest prime. However, we can create a bigger prime p(1) * p(2) * p(3) … * p(n-1) * p(n) + 1. This number cannot be a factor of any of the preceding primes, therefore it is a prime bigger than p(n). Thus, we cannot have a largest prime.
ultrafilter
Suppose p is the largest prime. Let’s construct an integer
N = 3 x 5 x 7 x 11 x … x p + 1.
Clearly N is larger than p and is not divisible by p or any prime lesser than p.
So, either N is a prime or is the product of two or more primes greater than p.
One thought experiment I like about infinity doesn’t even require infinite universes. Assume, for instance that the universe is neither going to expand infinitely or contract to nothing. Now, if you plot the eventual course of the universe, all we get is a dismal heat death due to entrophy. However, what happens AFTER is interesting. Remeber, entrophy is merely a statistical process, all it says is that there is overwhelmingly larger chance for order to turn to disorder than disorder to order. However, once were past the death of the universe, we have an INFINITE amount of time. As I pointed out in a previous post, we could envision, theoretically, random atoms bumping into each other in JUST the right way to spontaneously create every single SMDB member complete with all their memories and emotions at this very moment all sitting in front of working computers and then promptly suffocating from lack of air, all due to chance interactions. The sheer improbability of it is mindboggling give it would take some 10^10 seconds roughly to even get an electron to interact with something but we literally have all the time in the world.
Furthermore, and heres the really elegant bit, we could envision all atoms spontaneously converging on ONE exact point of space and exploding in the exact manner of the big bang. Not only would it be possible, but it would be inevitable. We might be the result of a previous universe 10^10^10^10 years ago.
You guys are missing the point. Yes, there is no largest prime. Like everyone else, I learned that back in high school.
What I am saying is that the notion of a largest prime has a rigorous definition, which Sérgio seems to have denied. I supplied a rigorous definition for the largest prime, which is something that we must have if we are to prove that no such thing exists.
Does anyone dispute that the largest prime is rigorously defined?
I’m going to sleep on this one 
Try not to dream about infinite identicle worlds, it gets complex, and you can never tell where you are when you wake up
My point is that the fact that we are talking about something doesn’t mean it exists.
We have a mapping from the power set to [0,1], that is true. But we should not be fooled into thinking that that mapping preserves all important inforation; simply because events A and B both get mapped to the same “probability” does not mean that A and B both share any essential nature; A and B can both have “probability” 1, even though A is logically necessary and B is not.
This is all true, but I really don’t see what you’re trying to say. Do you want a refineed notion of a probability distribution?
You are right. I misunderstood your previous post.
OK,THE UNIVERSE IS FINITE…OR NOT.
SAY IM STIRRING MY CUP OF COFFEE,AND TRYING TO WORK OUT ALL THE EVENT POSSIBILITIES OF EACH AND EVERY ATOM INSIDE THE CUP,TAKING INTO ACCOUNT THE FACT THAT ITS REVOLVING CLOCKWISE,AND THE RESISTANCE OF THE LIQUID AGAINST THE CUP,THE SPIN OF THE EARTH AND THE PULL OF THE MOON,…AND JUST WHEN ITS ALL WORKED OUT,GUESS WHAT…I FORGOT TO PUT IN THE SUGAR,…NEVER MIND,THAT WAS ALSO A THEORETICAL POSSIBILITY…START AGAIN…MUST MAKE DISCOVERY…WIN NOBEL PRIZE…COME ON PEOPLE,NEVER MIND THE COFFEE CUP…WHOSE DOING THE STIRRING?
I suppose so. If you’ve talking about an abstract mathematical concept, then it’s okay to say that something with probability 1 might not happen. But in the common use of the term, it wouldn’t have a probability exactly equal to 1.
in the common uses of probability (say, game theory), we are dealing with finite sets. if there are a finite number of possibilities, and one of the possibilities has a probability of 1, everything else must have a probability of 0 and the element that has probability 1 is guaranteed. when we deal with infinity (admittedly, “an abstract mathematical concept”), the rules change.
in an infinite set, an event with probability 1 is not guaranteed to happen. nor is an event with probability 0 guaranteed not to happen. it doesn’t seem intuitive, but that’s because your intuition is based solely on dealing with finite sets of events. that’s just the way it is.
This is all interesting reading. To return to the OP
It is undoubtedly easier to discuss the infinte than the infintisimal. So in this case it is easier to discuss an infinite number of possiblities than it is to discuss infintisimal probabilities. Math buffs with fine-tuned semantic arguments can do the translation.
Meta-Gumble hit the nail on the head with his OP. Just because there is an infinite amount of time to play with (and that is an assumption too), it does not necessarily follow that every possible configuration of the universe will occur.
It would take an infinite amount of time to poke a pin into every possible position on a postage stamp. The number of possible configurations of the entire universe is infinitely greater than this. Consider a universe exactly identical to the one we are in but with one of the atoms of my thumbnail relocated to Alpha Centauri. You get the idea. Of course there are infinitely more possible configurations again if you consider that the universe might itself be infinite.
Douglas Adam’s vision of a universe populated by mattresses (floomping around as I recall) is just a fantastic myth.
Whether the probability is absolutely zero because it is impossible, or practically zero because of the continuous nature of the probability distribution is rather academic and has been explained by better brains than mine. (Thanx Ramanujan and others.)
The answer to the common assumption that “because the universe is so huge, there just has to be another earth out there” is that it just isn’t true.
The idea that “infinite” means “everything must happen” is a logical (and incorrect) conclusion to frequentist interpretation of probability. I think Bayesian methods may avoid this, but I’ve just started formal study of Bayesian statistics.
Given either an infinite number of atoms or an infinite amount of time then the probability of a given “everyday” event occuring would tend towards certainty.
However, current cosmology suggests neither is the case. The number of atoms is finite, and the time during which an “everday” (or even interesting) event may occur is similarly finite if the universe expands unto heat death.
I’m not arguing on the basis of intuition, but on personal opinion. That things with probability one always happen is a convention, so saying that someone with probability one doesn’t have to happen isn’t necessarily false. But when you say that two things have the same probability, yet we have different expectations as to whether they will occur (virtually certain versus absolutely certain), I think that you’re stretching the meaning of probability. You’re turning it into an abstact mathematical concept, rather than a measure of some real worl property.
Yes, that’s the point. The real world doesn’t have infinite sample spaces, and you’ll find that in a finite sample space, E has probability 1 iff E is guaranteed to happen.
Is there any other way to do it?
Take a simple example. Suppose there’s a completely unkown physical constant between 0 and 1. There are some things we can deduce about the universe if it is in certain ranges. Is there a way to assign probabilities to these?
You can say ‘No can do. Sorry.’ but if you actually want to answer questions like the OP, I think it’s necessary to assume some notion of probability, and I can’t think of a useful one that would work in this case without having 0-prob ‘possible’ outcomes.
What would you have us do?
With the standard definition of a probability function, there ain’t nothing you can do about it.
However (and I’m no probability theorist), I imagine you could extend the notion of a probability function P. Instead of having P map into the real interval [0,1], have P map into the hyperreal interval [0,1] (or some other extension of the reals that includes infinitesimals). The idea, of course, being that sets which previously got mapped to zero, yet were still “possible”, would get mapped to mapped to an infinitesimal, while sets that were previously mapped to one, yet not “certain”, would get mapped to a number infinitesimally less than one.
In fact, I would be damn surprised if this hasn’t been researched before. Of course, it may be possible that doing this would force you to give up certain other “nice” properties of a standard probability function, but it’s not clear to me that that would necessarily be the case.
Probability is a mathematical concept! We use this mathematical concept to model some properties of the real world.