What does the term 'infinitely rare' mean in the context of theoretical physics?

[Fanghur]

Lately I have been having a quite vitriolic debate with someone on Sodahead about whether William Lane Craig's 'Kalam Cosmological Argument' is sound. I don't think that it is, and I think that it has been debunked on numerous occasions, but I digress. He continually cites the so-called 'BVG theorem', claiming that it absolutely proves that the universe had an absolute beginning; I am well aware that this is not the case and that the BVG theorem merely proves that the expansion of the universe had a beginning, as Vilenkin and Guth have both gone on record to clarify.

 

Can anyone tell me what this means exactly; this is a quote from Vilenkin that my opponent is continuing to throw at me because Vilenkin is speaking of 'infinitely rare' particles, which he then claims is equivalent to saying that they have zero probability of actually existing. Can someone please try and clarify this issue?

 

"I would say this is basically correct, except the words “absolute beginning” do raise some red flags. The theorem says that if the universe is everywhere expanding (on average), then the histories of most particles cannot be extended to the infinite past. In other words, if we follow the trajectory of some particle to the past, we inevitably come to a point where the assumption of the theorem breaks down—that is, where the universe is no longer expanding. This is true for all particles, except perhaps a set of measure zero. In other words, there may be some (infinitely rare) particles whose histories are infinitely long."

 

I don't think that Vilenkin would have bothered mentioning it if that literally meant that there is zero probability of them actually existing, but I don't know what he means by it.

[Griffon]

I'm sort of thinking out loud here, without knowing anything about the debate to which you refer ...

 

What does "infinitely rare" mean? Crudely put I would have thought that the chance of finding an infinitely rare particle amongst the infinite set of all particles is one over infinity - so damn near zero it might as well be. On the other hand, you just might get lucky!

[Fanghur]

Here is the link to the entire debate, if someone want to see it to better understand where I am coming from. If anyone wants to comment on whether they think this guy is just trolling or whether he is being serious, I would be interested in knowing that as well.

 

http://www.sodahead.com/entertainment/who-is-the-greatest-youtube-atheist/question-3595819/comment-106038411/

[ajb]

'infinitely rare'

= "infinitesimally small probability".

 

Statistically this is equivalent to zero probability, that is will never happen.

 

Informally, if something has a very small probability of occurring then you will have to wait a long time to see it, that is watch many events. If the probability is small enough, then you could have to wait longer than the Universe is old to have a reasonable chance of seeing this event. Such an event would practically not exist, but there is a small chance that it can happen or has happened.

 

This is what I think they mean, very small, non-zero but finite probability.

[Griffon]

Yes. Not "equivalent to zero" though. Isn't there an infinitessimally small probability of pulling a green ball out of a bag containing an infinite number of red balls? But you still might get lucky!

Edited March 30, 2013 by Griffon

[John Cuthber]

Ask the person who is using the phrase to define it.
If they can't explain exactly what it means, they can't use it as part of their argument.

[Fanghur]

It wasn't the guy who I was arguing with who said it, what he was trying to argue was that the Borde-Vilenkin-Guth theorem proved that the universe had an absolute beginning, therefore god must exist. Obviously that is a complete non sequitur, and the Borde-Vilenkin-Guth theorem only proves that the expansion of the universe had a beginning, so I checked up on it and it turns out the some of the people who derived the theorem don't even agree with my opponent's (or William Lane Craig's, who he is basically parroting) position regarding the theorem, and one of them had said something along the lines of "there may be some (infinitely rare) particles whose histories are infinitely long."

 

This was after he had clarified that the theorem doesn't mean that the universe had an absolute beginning, which was the only point I had been trying to make, but my opponent decided to latch onto the 'infinitely rare' thing, most likely because he knew that I had defeated his argument some just started flailing.

[Griffon]

Why is it described as a theorem, which is a mathematical term, rather than a theory?

[ajb]

Yes. Not "equivalent to zero" though.

I don't know what one really means by infinitesimally small in this context, probabilities are real numbers that lie between 0 and 1 (we normalise them that way). As infinitesimals are not real numbers we have some difficulty in understanding this. We say that statistically this is equal to zero. Or we are considering a continuous probability distribution and will be looking at events falling within an interval defining the probability via an integral. Each individual event may have infinitesimal probability, but over a small interval this can be finite.