So are you saying that something we know nothing about is more likely to occur more than once than it is to occur merely once, simply because {1} is a smaller set than {n∈N:n≠1}?
I was halfway through a long writeup.. and.. I just lost it all..
I'm typing it out again, bu tI'm rushing through it, sorry, it's going to be a bit less coherent than the first time around:
Let P(X) = the probability that life arises on a planet from the beginning of time til now
q = the number of planets in the universe
A. We have no idea what P(X) is.. It could be 0.7, 0.0000007, something like that with 200 zeroes, googol zeroes, it could be 7%, or 28%, or 80%, or whatever.
B. We have no idea what q is, it could be anything from some large number to a much larger number.. or maybe even infinity.
C. To get the expected number of intelligent life which has arisen in the universe so far you multiply P(X)*q = E(X).
D. P(E(X)=n) is the probability that the expected value of number of intelligent civilizations which have arisen is n.
E. Because of A, B, C, and D, we can't at all say what P(E(X)=1) is or P(E(X)=2) is, or P(E(X)=n), for any n from 1 to q is.
F. Therefore there is not enough data to say that P(E(X)=1) is a special case and is more likely than all the other possible scenarios. No case is special. The only thing you can do is sort of estimate how many planets there are in the universe, and that narrows things down a bit, but in the end P(X) is completely unknown, so that doesn't change much. And okay, P(X) is probably not 80%, but the point is that it's an incredibly wide range of possibilities.
This took like 2 hours
I hope it makes sense. If not, I might need a couple weeks to get back to you.
Proper form is not necessary. The key is the distinction between a large but finite number and a true infinity. If you assume an infinite universe. It follows that the probability an exerimental result of one and only one is infinitely small. That is not hard to follow.
That the argument works in the case of infinity is only a part of it. It does break down on some sort of a level of scale for sure.. It's easy to imagine several scenarios. We can't say at which scale it breaks down though, whether it's on the scale of the arm of a galaxy, or a galaxy cluster, or whatever. The point is that at a large enough scale the argument works. Whether that scale is larger than the size of the universe is irrelevant to the argument, I think, because we just don't know much about P(X) or many other things. To get exactly 1, and for that to be more likely than everything else, a lot of stuff just has to line up right. We don't know squat, so we can't make assumptions, is basically what I'm trying to say.
There are issues. For example, certain physical constants need to remain constant. In an infinte universe, that is not a given. We live a fairly flat space curvature. Is that important? Many potential factors are simply unknown.
I think these issues help my argument. It sort of plays into my analysis of the situation above I think, by highlighting even more how much we just don't know. We don't know what sort of life is possible, so you can't say that the probability of life in a gas giant is this or that, or on a methane planet, or even things we can't imagine. It exposes how many crazy variables there are in this equation and how much variance there could be in the data.
I say again, it is not unreasonable to assume we are alone.
Whether it's unreasonable is beside the point I think, I'm just talking about whether it's likely or not. Unlikely things can still be reasonable. The crux of my argument is that we have no idea how likely or unlikely, so I can't say it's not reasonable, if I'm going to be fair. So... unlikely but reasonable. I guess that's what I'm settling on.
