Isn’t using the example in my OP enough to at least implicitly define them (even if I, not really recognizing the need, didn’t do so explicitly)? For instance, ‘C can happen’ would mean ‘there are worlds (in the sense of sequences of the events A, B, and C) that include C’, or is that not a workable definition (and to some extent suggested by the example)?
Perhaps I would’ve done that, I just never heard of possibility theory before I wiki’d it just now, or perhaps I just forgot. However, the wiki article certainly seems to insinuate that there is a correspondence between possibility and probability whether or not we impose one, and given the intuitive meaning of the words I would expect (not that this expectation needs to be born out) that at least what is probable is also possible, if perhaps not the other way around (as shown by things with zero probability happening; perhaps the assumption that this works the other way round, too, was what led me to my confusion in the first place). Is that wrong?
The examples I gave in post #38 were intended to show that, no, your definition is not as workable as I’d like it to be. Specifically, if “C can happen”, does that mean that it is definitionally capable of happening, or that it is practically capable of happening? Definitionally, there’s nothing to prevent a citizen of Antarctica from being the first person on the moon. But practically speaking, someone else got there first, and Antarctica doesn’t have citzens anyway. So I don’t know whether you would consider this to be something that “can happen”.
Perhaps we have a semantic misunderstanding. Are you saying that if an event has a zero probability, then it might happen anyway? My understanding is that “zero probability” = “cannot happen”. Maybe this is the whole problem in why we don’t understand each other. Can you go into more detail about how something with a zero probability might happen?
That last one is possible. There’s a qualititative difference between things that are logically or definitionally impossible (like pentagons with four sides) and things that are just absurdly unlikely. Rocks spontaneously writing DoI is a perfect example of something that should happen, giving infinite time.
“zero probability” = “cannot happen” only in situations where you have a finite sample space (i.e. there are only finitely many things that could happen).
In order to study probability in a mathematical, logically consistent way, you have to have some axioms, like the ones ultrafilter linked to in Post #15. Those axioms assert that if you add up the probabilities of all the different (mutually exclusive) ways something could happen, the total has to be 1 (or 100%). This would be impossible if you had infinitely many outcomes, each with a nonzero probability.
Here’s an example. Consider throwing darts at a dart board in such a way that each point on the board is equally likely to be hit. (Note that this assumption takes us into the realm of mathematical abstraction, not real-life darts-playing.) We can say that the probability of hitting any particular section or region of the dartboard is equal to the area of that section divided by the total area of the board. I think this is consistent both with the commonly accepted axioms of probability and with our intuition.
So if the dartboard were divided in half, there’s a probability of 1/2 (or 50%) for either one of the halves. If the board is divided into 20 numbered sectors of equal size, the probability of the dart landing in, say, the 13 sector is 1/20 (or 5%).
Since a point has zero area, the probability, as defined above, of a dart hitting any particular point is 0. (So is the probability of the dart hitting one of a finite set of points, or of hitting any particular line.) So the dart has zero probability of hitting any particular point, including the one you just did hit when you threw it, even though it’s not impossible for the dart to hit that point.
How this relates to the OP is that the set of all possible infinitely long strings made up of A’s, B’s, and/or C’s is an infinite set, just like the set of points on the dartboard. Choosing one of those strings is like hitting one point on the board. Strings that have no C’s in them (i.e. made up of only A’s and B’s) do exist, so it is possible to choose one of them. But if the set of such strings has an “area” of 0, in some sense (which gets into what’s called “measure theory”), there would be zero probability of selecting such a string.
This sounds so much like Zeno’s paradox that I’m getting dizzy just thinking about it.
A point has zero area only in the theoretical world of math. You can’t extend that idea into the real world. That’s what got Zeno into trouble. When we say that a point has zero area, what it really means is that it has an infinitely small area. Too small to be measured by any real-world instrument, but it is sufficiently non-zero to allow calculus to work. If it were truly zero, calculus would fall apart.
If you want to play with infinities, this is how I see it: Your dart board has an infinite number of points on it. Each one is infinitely small. If you would repeatedly throw your dart for an infinite amount of time, you WILL hit every point. Given enough time, anything that can happen, will happen. You’ll ask, “Ah, but there’s another point between those two points!”, and I will answer, “No problem - we still have an infinite amount of time ahead of us!” If there is a point which still did not get hit even after an infinite amount of time has passed, the only reason why is that it couldn’t get hit.
Sure, it was fine; that definition amounts to putting no special constraints on possibility at all. “Those things are possible which I say are; a ‘possibility distribution’ is just a set of possible worlds”. And in the case of your problem, if you take the set of possible worlds to be all infinite sequences of letters, it makes it very clear: it’s possible to have an infinite sequence with no Cs, even though, for any finite string, it’s always possible that C will come next. “Anything that can happen, will happen, given enough time” is falsified by that possibility distribution, on that minimal account of what possibility is.
Oh, geez, I just made up the words “possibility theory” to speak informally; I didn’t realize Zadeh had already taken it for one of his fuzzy logic things. Sorry, I wasn’t referring to what’s on the wiki article!
Even then, they aren’t equal. There’s nothing stopping you from saying “This coin has a 100% probability of coming up heads and a 0% probability of coming up tails; still, it might come up tails all the same”. (After all, you could always extend this with the explanation that, behind the scenes, God throws a dart at the dartboard; if he hits the exact center, tails, otherwise, heads. Both are possible, but one has 100% probability. Not that you need to invoke such an explanation; there’s nothing stopping you from giving 0% probability to an event which is possible, even in a finite sample space, unless you go ahead and impose that restriction yourself. There’s no particular link between probability and possibility.).
And thus each particular sequence has probability zero, is Thudlow’s point (at least, as long as they all have to be equiprobable). So that a probability of zero doesn’t mean “guaranteed not to happen”. It just means “a probability of zero”. There’s no particular link between probability and possibility.
You can go ahead and try to make this intuition work in some formal system; more power to you. But understand that, when mathematicians standardly employ terms like “A point has an area of 0”, they aren’t being wrong; they’re just using a different formalization of the notion of “area” than the one you’re proposing, a formalization in terms of which “has an area of 0” does not mean “is empty”. And why not?
Alas, there are different levels of infinity. If you just sit around throwing darts for a countably infinite long time (that is, once for each natural number), as has been the assumption throughout this thread about what doing something infinitely often means, then you can’t possibly hit all uncountably many points on the dartboard. You can’t even hit all the points on one particular line segment on it (suitably mathematically idealized). See the proof here.
No, calculus is defined in the theoretical world of math, where points do have zero area.
And this is demonstrably false (as proven by Cantor). The sequence of points you hit by repeatedly throwing your dart is countably infinite (cardinality aleph-null), while the set of points that make up the dart board is uncountably infinite (cardinality c). No matter what infinite sequence of points you hit, there will always be points on the board that are not part of that sequence.
It may be that the set of things that can happen is of a higher cardinality than the set of things that will happen, given infinite time. If so, it follows that there are things that can happen that won’t happen.
I know, it sounds like a contradiction in terms to say that something can happen but won’t happen. Whether it is or not depends on how we define our terms.
First, I would love to know what you mean by “countably infinite”. I thought that “infinite” means so big that it can’t be counted.
Second, is the case of the dartboard really different than taking a curve, dividing the space under it into an infinitely large number of infinitely narrow strips, and multiplying them to get a finite area?
Incidentally, nothing in calculus demands that points have non-zero area. It just demands that things which have infinitesimal area have infinitesimal area. It doesn’t demand that those things be single points…
I’ve clarified above what “countably infinite” means. It means you can assign one to each natural number, in such a way that you cover them all.
I have no idea what you’re referring to with multiplying an infinitely large number of infinitely narrow strips. I assume you mean adding up their area instead (to integrate a function)?. Which is indeed the same thing; if you take each particular strip to be only 1 point wide, then, on the standard account, each strip has area 0, even though the whole thing has non-zero area; what of it? On the standard account, you can break a non-zero area up into an uncountable number of regions of 0 area with no problem.
Of course, if you take each strip to be a little more than 1 point wide (let’s call this an infinitesimal amount), then it has some infinitesimal area, but that’s not in contradiction to the above.
This is like the old “one thousand monkeys at one thousand typewriters will eventually produce all the works of Shakespeare” isnt it? Might take a while, especially with only a thousand monkeys, but there is no mathematical reason it wouldn’t happen. Or as The Simpsons put it:
Mr. Burns: “In this room I have a thousand monkeys typing away at a thousand typewriters. Soon they will produce the greatest novel known to man!”
(Mr. Burns picks up a piece of paper from a typewriter)
Burns: “It was the best of times, it was the BLURST of times! You stupid monkey!”
I know there are a lot of programmers on the board. Has anyone ever written a small program to see just exactly how long would it take for a random sentence generator to exactly produce “It was the best of times, it was the worst of times”? Just curious, I think it would be really interesting to know
Well, the amount of time it takes is random, of course. If you keep spitting out random messages of length L constructed in an alphabet with K many characters, and even have a mechanism which prevents repeats, it will take, on average, (K^L-1)/2 wrong attempts before you get it right. Even removing spaces and the comma from your message, and ignoring capitalization, that’s a message of length 39, in an alphabet of size 26; so, on average, you’d take about 7.6 * 10^54 attempts. How long that is depends on how fast you keep spitting out messages, but it’s one of those mind-destroyingly large numbers that means “Forget it; neither you nor any of your descendants will ever see this happen, in all massively overwhelming likeliness”.
It has been attempted, but I don’t think it has ever been done, considering that there are on the order of 10^50 possible combinations.
However, it should be noted in passing that if you are allowed to keep the “correct” characters with every iteration, you can get the phrase in less than a million (perhaps less than 1000!) iterations. Something to remember when talking about randomness and evolution, which I know we’re not.
I skimmed that article. And I skimmed Wiki’s articles on “Aleph number” and “Countable set”. And I now say with a fair amount of confidence that I am not competent to continue this discussion.
I hope that someday I will understand the difference between the countably infinite and the uncountably infinite. But that day is not today. For today, it will have to suffice that I recognize those who have studied these things more than I have, and that there is a difference.
All of this is quite sensitive to exactly what mathematical formalization one is working with. After all, who says the points on a dartboard correspond to the standard construction of the real numbers? Even taking the position that they do, that construction and its properties are in turn sensitive to ambient details of the set theory one is working in. For example, in intuitionistic set theories, there can consistently be a surjection from the natural numbers to the real unit interval, even though Cantor’s diagonalization proof is also valid (it just happens to prove something else).