[omnicognate]

Strictly speaking our own existence doesn't even tell us the probability is greater than zero. For an event to have probability zero doesn't imply it can't occur. If a number is chosen from a uniform distribution on the reals between zero and one, whatever the result is the probability of that exact result occurring was zero.

[dotancohen]

  > For an event to have probability zero doesn't imply it can't occur.

The distinction in meaningless. We exist, ergo intelligent life can develop in this universe.

[omnicognate]

> The distinction in meaningless. We exist, ergo intelligent life can develop in this universe.

I didn't say anything contrary to this. I was just pointing out an interesting detail about probability theory.

It's impolite to edit your comment in such a way as to turn its existing replies into non-sequiturs. For the record, this comment initially cast doubt on the claim about zero probability events, hence the reply saying it was correct.

[danbruc]

It is correct, if you split 100% across infinitely many possible outcomes, each outcome will have probability zero, still one of the possible outcomes will occur.

[rsj_hn]

Doesn't this require a continuum? I don't think you get there with a measure on a countable set. That would make this an unphysical argument, IMO.

[omnicognate]

Yes, it requires uncountability. It could be that physics is ultimately best modelled with countable sets, but that hasn't been established and our current best physical theories are certainly full of uncountability.

The distinction between "surely" and "almost surely" [1] is "just" a curiosity about probability theory though, albeit a rather fundamental one, and I only brought it up as such. It's interesting to think about, and if you do so it quickly brings you up against deep philosophical questions about what probability means.

[1] https://en.m.wikipedia.org/wiki/Almost_surely

[danbruc]

Just out of curiosity, is there are short answer why it requires uncountability? Naively something like pick a random natural number would also seem to lead to probability zero. I can see that pick a random natural number might be problematic, how would you do this? Pick on digit and then with some probability either stop or continue and pick another digit, but it is at the very least not obvious that one could make this work without larger numbers just having smaller and smaller probabilities and there might also be issues with termination. On the other hand it is not obvious to me why one could not work with uniform distributions over the set [0, n) and then look at the limit as n goes to infinity.

[rsj_hn]

If you have a measure on a countable set, lets number it 0, 1, 2, .. then you must have: m(i) >= 0 (since it's a measure).

And must also have

1 = m(0) + m(1) + ... (because it's a measure)

so

1 = Lim S(i)

Where S(i) is the partial sum going from 0 to i.

But if each m(i) = 0, then each partial sum is zero.

So 1 = Lim 0 = 0

[rsj_hn]

> It could be that physics is ultimately best modelled with countable sets

That's not my argument. The issue isn't whether physics requires a continuum to model reality -- I'm certain it does. But just because a continuum is required to model the universe doesn't mean that the observables in the universe actually form a continuum. For that, I am certain that they don't.

[danbruc]

It seems to me, the question is, how do we assign probabilities to the existence of life. One way I can imagine is the following. We think of the universe as a classical system, then there is a phase space for the entire universe. Now we can look at each trajectory through phase space and classify it as either having or not having life at at least one point. Then we can obtain the measure of the set of trajectories classified as having life. With this view it seems at least possible that life could have measure zero even though it does not seem likely to me and there might even be [non-]obvious reasons why the set could not have measure zero. I am not sure how the argument would change if one would try something similar but with a quantum mechanical instead of a classical description of the universe.

EDIT: Additional thought and I might be totally wrong because of a lack of mathematical understanding. Pick a point on a trajectory classified as containing life and perturb it in a way such that it only affects parts of the universe far away from life. Then all trajectories through the perturbed points would also still be classified as containing life. But I think the resulting set of trajectories would still have measure zero because we allowed only perturbation far away from life.

So to grow a single trajectory classified as containing life into a set of trajectories classified as containing life of non-zero measure would require being able to pick a point on the trajectory and perturb it in all dimensions and still have all perturbed trajectories classified as containing life. Seems possible but not obviously so to me.

[mensetmanusman]

Infinity times zero might be meaningless in a quantized reality