To get an infinite set of texts with a finite set of characters, you need texts of infinite length. I think it is similar for dreams—the set of possible experiences is finite, and dreams have a finite sequence of experiences.
The pool of possible dreams is so large that we will never hit any limit—and even if (which would require experienced lifetimes of 10^whatever years), we would have forgotten earlier dreams long ago.
You get an infinite set of texts with a finite set of characters and texts of finite length merely by letting the lengths be unbounded. Proof: Consider the set of characters {a}, which has but a single character. We are restricted to the following
texts: a, aa, aaa, aaaa, aaaaa,… We nevertheless spot an obvious bijection to the positive integers. (Just count the ’a’s) So there are infinitely many texts.
Sorry, I was a bit unprecise. “You need texts without size limit” would be correct. The issue is: Your memory (and probably lifetime) is finite. Even if you convert the whole observable universe to your extended memory.
But outside an infinitesimally small subset, so tiny that any attempt to express it as a fraction would just give you zero, you couldn’t appreciate any of the texts within a human lifetime, even if you managed to get clever and extend the lifetime to the heat death of the universe.
I do wonder about a culture running out of ideas, high concepts, that sort of thing. Dreams can be long and messy, so the permutation space of the limited-by-finite-lifespan-of-physically-embodied-agents set of dreams is still huge. Good ideas, on the other hand, are often things you can distill to a short sentence in ordinary language. You can describe an interesting idea in ten common words. Let’s say it takes a day on average to evaluate whether any one idea is good or not and that there are ten thousand common words. There are 10000^10 = 1e40 such sentences, a small minority of which will describe coherent ideas.
It’s seems quite physically possible to have a civilization last several millions of years. This would give the civilization a total of 1e10 days. That’s 1e30 ideas to think about each day. A good galactic civilization should be able to colonize all of the Milky Way, giving it something in the excess of 1e10 stars to build habitats around. An average Dyson sphere built around a star populated by 1e20 people gets a population density of around 500 people per square kilometer, around the same density as in the Netherlands.
So all you’d need is a Dutch galactic supercivilization spanning some millions of years and really, really obsessed with word permutations to utterly exhaust the ideas expressible in ten common words. Anything interesting they won’t have thought reasonably carefully about will be literally inexpressible in ten words, unless you start expanding the language with new words.
And compared to sets of strings with unbounded length, there is nothing particularly outlandish about those numbers. Science routinely handles far larger orders of magnitude of both time and space.
That’s just if we’re operating in a 1950s future paradigm, where you need to do everything with regular humans running around and occasionally colonizing neighboring solar systems when things get crowded.
If we’re allowed to do a bit of virtualization, then things get more interesting. We could get things a bit more compact if we take a few centuries at the start of the project to develop solid brain emulation technology to ease those pesky problems of needing lots of living space, sleep, and eventually devolving into stone-age cannibalism with mystery cults around permuting ten word sentences. Estimating the computation involved in human cognition is tricky, but 1 exaflops is floating around. Say that a well-engineered and focused emulated mind can evaluate one permutation in an average 1e4 seconds, a bit less than three hours, since it doesn’t have to worry that much about maintaining a society.
So that means the exhaustion process would require 1e18 1e4 1e40 = 1e62 computation steps. A kilogram of Drexlerian nanocomputers appears to be able to do something around 1e25 flops.
So if you were in a big hurry and wanted all simple new ideas ruined in just 300 years, you could just grab Jupiter, turn all of it into drextech computronium, and fill it with your loyal EM programs. A technologically advanced Kardashev II civilization might end up having exhausted a lot of simple ideaspace after a single millennia of hanging around in a single solar system.
To get an infinite set of texts with a finite set of characters, you need texts of infinite length. I think it is similar for dreams—the set of possible experiences is finite, and dreams have a finite sequence of experiences.
The pool of possible dreams is so large that we will never hit any limit—and even if (which would require experienced lifetimes of 10^whatever years), we would have forgotten earlier dreams long ago.
You get an infinite set of texts with a finite set of characters and texts of finite length merely by letting the lengths be unbounded. Proof: Consider the set of characters {a}, which has but a single character. We are restricted to the following texts: a, aa, aaa, aaaa, aaaaa,… We nevertheless spot an obvious bijection to the positive integers. (Just count the ’a’s) So there are infinitely many texts.
Sorry, I was a bit unprecise. “You need texts without size limit” would be correct. The issue is: Your memory (and probably lifetime) is finite. Even if you convert the whole observable universe to your extended memory.
But outside an infinitesimally small subset, so tiny that any attempt to express it as a fraction would just give you zero, you couldn’t appreciate any of the texts within a human lifetime, even if you managed to get clever and extend the lifetime to the heat death of the universe.
I do wonder about a culture running out of ideas, high concepts, that sort of thing. Dreams can be long and messy, so the permutation space of the limited-by-finite-lifespan-of-physically-embodied-agents set of dreams is still huge. Good ideas, on the other hand, are often things you can distill to a short sentence in ordinary language. You can describe an interesting idea in ten common words. Let’s say it takes a day on average to evaluate whether any one idea is good or not and that there are ten thousand common words. There are 10000^10 = 1e40 such sentences, a small minority of which will describe coherent ideas.
It’s seems quite physically possible to have a civilization last several millions of years. This would give the civilization a total of 1e10 days. That’s 1e30 ideas to think about each day. A good galactic civilization should be able to colonize all of the Milky Way, giving it something in the excess of 1e10 stars to build habitats around. An average Dyson sphere built around a star populated by 1e20 people gets a population density of around 500 people per square kilometer, around the same density as in the Netherlands.
So all you’d need is a Dutch galactic supercivilization spanning some millions of years and really, really obsessed with word permutations to utterly exhaust the ideas expressible in ten common words. Anything interesting they won’t have thought reasonably carefully about will be literally inexpressible in ten words, unless you start expanding the language with new words.
And compared to sets of strings with unbounded length, there is nothing particularly outlandish about those numbers. Science routinely handles far larger orders of magnitude of both time and space.
Quite the caveat.
That’s just if we’re operating in a 1950s future paradigm, where you need to do everything with regular humans running around and occasionally colonizing neighboring solar systems when things get crowded.
If we’re allowed to do a bit of virtualization, then things get more interesting. We could get things a bit more compact if we take a few centuries at the start of the project to develop solid brain emulation technology to ease those pesky problems of needing lots of living space, sleep, and eventually devolving into stone-age cannibalism with mystery cults around permuting ten word sentences. Estimating the computation involved in human cognition is tricky, but 1 exaflops is floating around. Say that a well-engineered and focused emulated mind can evaluate one permutation in an average 1e4 seconds, a bit less than three hours, since it doesn’t have to worry that much about maintaining a society.
So that means the exhaustion process would require 1e18 1e4 1e40 = 1e62 computation steps. A kilogram of Drexlerian nanocomputers appears to be able to do something around 1e25 flops.
So if you were in a big hurry and wanted all simple new ideas ruined in just 300 years, you could just grab Jupiter, turn all of it into drextech computronium, and fill it with your loyal EM programs. A technologically advanced Kardashev II civilization might end up having exhausted a lot of simple ideaspace after a single millennia of hanging around in a single solar system.