Many of my readers are intelligent and scientifically-literate people. So today I’d like to share this cute paradox I came up with during my vacation last month. I was discussing some computer science topics with my buddy Luqman when I got this idea.
The paradox itself not that hard to resolve; one can argue that it’s not an actual paradox but rather a layman’s misunderstanding of the nature of “information” (I’m not surprised if someone has thought of this before). The paradox is very simple, and this is how it works: a seemingly reasonable hypothesis results in an absurd conclusion, so there must be a logical error somewhere. Your job is to spot where the logical error occurs.
If you can resolve the paradox, please provide your explanations in the comments field below. I put this up on /sci/ and it took less than a minute for a forum member to solve it.
Consider all the photos you can take using a standard digital camera. Each photo is determined uniquely by the color of each pixel (other properties like brightness, contrast, etc are simply a product of pixel coloring and so are not relevant here).
Since the photo is digital, there is a finite number of pixels, say M (roughly in the order of 1 million), and each pixel can be colored in a finite number of colors, say N (assuming 8-bit color, we have 256 x 256 x 256 possible colors). We do not care about the actual values of M and N. The only important thing to keep in mind is that while M and N are huge numbers, they are finite.
By a basic counting argument, the number of all possible images is N^M (N raised to the power of M). A huge number, but still finite. We call this number K.
Now, all possible images that can be taken with the camera belong to this collection of K photos. Loosely speaking, this collection of K photos contains all possible images in the universe, at the resolution which the camera allows.
Suppose I’d like to perform this process: First, I write the number 1 on a paper and take a photo of that paper. Then I write the number 2 on a paper, and take a photo of that paper. Then I repeat this process, going on and on until I reach the number K+1 (it might take a very long, but still finite, time). Now I have generated K+1 photos. This is a contradiction to our earlier statement that there are only K possible photos that can be recorded with the camera.
Where does my argument break down?