By accident I stumbled upon this topic and I cannot resist to post a comment. Starting with the first post
A friend and I were discussing an idea he had which would cycle through every possible pixel combination to produce every possible image. Basically, it would treat the monitor like a giant number, whose base is the number of possible colors for each pixels. For instance, a 1280x1024 resolution monitor using 16mil color could represent a total of (16mil)^(1240*1024) images, which would probably take 2*infinite amount of time to render.
I suppose you mean "all possible images, that this monitor can produce", then the time it takes, no matter how long it takes to show one of the (16mil)^(1240*1024) possibilities, is finite. If you do one image per second it will take exactly (16mil)^(1240*1024) seconds.
If you mean "will you have then all possible images", the answer is :
noproof
imagine all (16mil)^(1240*1024) possible and different from each other, images, on a row. Take the first image. Draw a nice thin white, one pixel line around that picture. That picture is certainly different from all the pictures in the row.
Another proof:
This is a picture of zero
0Now we decide from now on we represent zero with another picture, the black screen from the monitor above. Then we create a picture for 1, by changing a pixel in color, then we creating 2, by chancing again the color from that pixel. Going on and on we can create images of all numbers up to (16mil)^(1240*1024) . But It is not possible to create an image for (16mil)^(1240*1024) +1.
This idea is interesting because, if there is a human being who can remember all the screens together with the number it represent, then, for him, every screen picture would have meaning.
Now, make a drawing. Will that drawing be in the row of all possible pictures of the monitor.
The answer is: no. More precise the probability that it will be there is, believe it or not, zero. (I will come back on this later)
Take a picture of your drawing with you digital camera, or take 1000 pictures. Render them on your monitor. Are they in the row from above. Sure. Because the monitor can show them, they must be in the row.
Now, sit down for the monitor. Let it show pictures, at random, out the row from above. Will you see one of the pictures of your drawing. The chance to see one is definitely not zero. But "Will you see one?"
Eh, no.
Let`s make the chance (probability) a bit bigger. Let us take all pictures rendered by the monitor that has a meaning for you, that you recognize as....... Will you see one of them? The probability that this happens is certainly millions and millions times bigger as the first probability. But will you recognize something?
Eh,eh no.
This has nothing to do with infinity. It has to do with big numbers and really big numbers. And the huge difference between them.
I cannot know how many screen captures of the monitor will have a meaning for you. But I am sure there will be no more than 10^157 such pictures. Why I take 10^157? Because I can connect it to something, that has some meaning for you: There are billions of galaxies, with billions of stars. billions of planets. On our earth there are billions and billions of molocules. Nevertheless the total number of all particles in the universe is smaller than 10^157. This is a big number. Bit not really big.
But can you image that there are actually more than the number of particles in the universe pictures stored in your brain, and having a meaning for you? It seems a save upper bound. Now what is the probability to see one of those pictures? It is 10^157/(16mil)^(1240*1024). And that is not a small, but a really small number. To make it even more clear. Imagine 10 billion worlds, with 10 billion people, all watching for 10 billion years to a monitor, with every second a picture, at random. out of the possible pictures of that screen. They will have seen about 10^36 pictures. The possibility that one person, on one of those planets, at one moment will see a picture with some meaning for him is about 10^36*10^157/(16mil)^(1240*1024). About 10^(7000000-193). Still a very small number.
Although the probability is not zero, for every practical purpose it is. Looking to a monitor, generating pictures ad random, you will see, for certain, only grey shaded, flickering images.
more clearly:
Everything that you ever have imagined, actually exists somewhere because the human brain (compared against the universe) is pretty limited and cannot create something that is out of the universe..
I don't think so. It is exactly the opposite. The number of somethings in the universe is finite. And although the number of everythings you have imagined is finite, your possibilities are infinite. And many, many things you have imagined don't exist in the universe. For instance all , I repeat, all math objects do not exist in the real world. Zero is not lying in the grass, 1 is not hanging in a tree. There are no straight lines, no perfect cubes in the universe, and no 10 dimensional sphere either. It may be sad, but math don't really exist.
Back to the probability of zero, the probability that the screen renders you self made picture. The space of pictures, the space where pictures "lives" is uncountable infinite. Now we have to make a difference between countable infinite, that is when you can put the elements of the space in an infinite row, and uncountable infinite if that is not possible. The natural numbers are countable, the real's aren't. If you pick out, ad random, a real, the probability that number is a natural one, or even a rational one, is zero. Quite a paradox. (The problem is, we do not have a mechanism to take ad random a real). Maybe this will clear up things a bit: The surface of the line x=1 is zero. The surface of all lines x=1, x=2, x=3, ......(that are countable many lines) is also zero. Nevertheless the surface of all lines x=a, x between 0 and 1 is.....infinite.
In an uncountable infinite space the probability that something out of a countable set of elements of that space happens is zero.
Some people think that all images are in the Mandelbrot set.
I don't think so.
There are uncountable many images in that set. But that is not "all images".
I think that there are no straight lines or a square in the set. There must be a mathematician that can prove such a thing.
Also the argument of Syntopia, by working around this , by coding, is not correct. You cannot code all images, an uncountable set, with for instance the countable many subsets of the decimals in PI.