I’ve snipped off the context that clarifies what you mean, but the wording above is terribly amusing to me.
Re: the OP:
One might well take “possible” to mean “Happens somewhere at some point”. On that account, everything possible happens somewhere at some point, tautologically…
If you do not want to use that definition of “possible”, well, I daresay, the burden is on you to first clarify what you mean by “possible” before we can tackle the question “Must everything possible happen somewhere at some point”? (While you are it, you may as well clarify what “Must” means, if you are interested in asking the question with a notion of “Must” less stringent than tautological necessity)
(Similar considerations apply to the use of probabilistic language rather than possibilistic language, if that’s what floats your boat.)
In statistical mechanics, a system that (eventually) takes on all of its possible states (to within some precision) is known as an ergodic system. Whether or not a given system is ergodic is not always easy to ascertain, though.
Okay, let’s take a specific example.
Pi has an infinite number of digits in a non-repeating sequence. Will every possible sequence of numbers occur there sooner or later?
Suppose you take the digits as text encoding. Will every conceivable message appear in the text somewhere?
This one? The entire text of Hamlet? The near complete text of Hamlet, but with one word omitted?
The answer to all these questions is “Almost everyone thinks ‘Yes’, but no one has ever proven it one way or another”.
I think this little back and forth is extremely interesting. Just a shout out.
But notice what this means, that it hasn’t been proven: it means that it’s not necessarily true just because the digits of pi go on forever (without repeating). We don’t (yet) know a reason why it couldn’t be false.
And that, to me, settles the OP’s question with a No. Since you can have an infinite string of digits without all substrings being inevitable, you can have an infinite stretch of time without all events being inevitable.
Just checking: we don’t know a reason why the digits of pi keep spooling on, or if there is a repetition “gotcha” moment?
Things, perhaps: should all things mutate by themselves or by some external influence such as unpredictable future events.
Concepts, probably not: Ideas such as mathematical concepts cannot because they are not things.
We know the digits of pi “keep spooling on” without ever just repeating the same sequence over and over; what we don’t know is whether the digits of pi contain every possible finite subsequence. That is, we know for sure that pi is irrational, but we only suspect (can’t prove) that it’s normal.
(Although it’s not all that obvious why pi is even irrational. It was relatively late in the history of mathematics, compared to when pi was known of, before anyone managed to prove its irrationality.)
Sure; for very simple example, the infinite string “00000…” never contains a 1.
But who knows whether that’s the sort of thing which will satisfy the OP.
The exact calculation would depend on the exact functional form of the evolution of the Universe, as well as on just which particular unlikely event you’re trying to find the probability of. Under any reasonable set of assumptions, though, the integral would converge very quickly.
Leo Bloom, it’s known with proven certainty that pi is irrational, which means we’re sure the digits won’t repeat. What we don’t know, but very strongly suspect, is that it’s normal, which means that all possible digit strings of any given length are equally abundant. And even if it’s not actually normal, it might still have the property that all digit strings show up in it. Then again, it’s also quite conceivable (for all we know) that there might be some digit strings which never show up in it. For comparison, consider the number 1.101001000100001000001000000100000001… (with an ever-increasing number of 0 digits between the 1 digits). This number is certainly irrational, since it never repeats, but it’s also certainly not normal, since you’ll never find the digit string “2” in it.
In infinite time, how can any possible thing happen, but only a finite number of times? After the rare event has happened a few thousand times, there is still an infinite duration remaining.
Did you perhaps mean to express this in terms of relative frequencies, rather than absolute totals?
It’s easy… think of a string like 101101001011010000000000000… [all 0s from thereon out]. 1 only shows up finitely many times.
Is there some reason a string like this doesn’t describe a possible way for eternity to go? (This is why I wanted the word “Must” clarified before…)
Well, OK, but that’s a non-normal scenario constructed specifically to achieve the outcome in question. Is there any evidence that the universe is like the string you describe?
Well you’ve ruined MY daY !
I think when you throw in the concept of multiple universes, this question is unanswerable.
Typically I’m not welcome into discussions like this, I’m venturing outside my intelligencia here. You dudes are *smart *(this thread makes me need a freakin’ nap.)
This number sequence helps guys like me make sense of the universe–Finite occurrences in an infinite string.
Just in practical terms, I imagine putting all the required fundamental base elements (minerals, metals, organic matter, etc) in a big bag.
Given an infinite chances to shake the bag up, I guess there’s a theoretical number model for the odds that all these base elements will congeal together in precisely the required formation for a working 1981 Pontiac Trans Am being shaken together. A black one. With leather trim, wood grain steering wheel and galaxy black paint with a gold flamin’ chicken on the hood.
There’s, I guess, a calculation for the odds shaking the bag up will yield that extremely specific result–but in practical terms, it simply wouldn’t ever happen.
EVER.
Would it…?
A theoretical number model’s potential result of, hypothetically, 1million gazillion:1 odds doesn’t at all mean that if you actually built a perpetual shaking-the-bag-up machine, it will ever, EVER happen…?
The question is more like: if you put a thousand raisins in a cake mix of infinite size, would they all sink to the bottom?
If an event is of sufficient rarity that it can only occur a finite number of times in an infinite period, that means the infinite period can be divided into two separate infinite periods, in one of which, the event doesn’t occur at all.
If we posit that all units have a certain set of characteristics, then all combinations are resolvable.
If all units are random, then maybe not.