Is there a mathematical justification/proof for the claim that the universe contains a finite amount of information? Since I was first introduced to it, I've been intrigued by the claim that the universe contains a finite amount of information. (That link is not where I first encountered the concept; it is simply the first example of this claim I could find from a quick Google search.)
Basically, the argument seems to be that if there is a finite amount of matter in the universe, that matter can only store a finite amount of information. On the surface, I have to concede that this makes a lot of sense. After all, if I'm thinking in terms of bits (for example), I might visualize a hypothetical "infinite hard disk drive" that could store unlimited data. This device would presumably have to be infinite in size, since it stores information on a physical platter that obviously occupies some space.
Digging a little deeper, however, I start to doubt this presumption. After all, information can be compressed according to a system of encoding information in a particular set of symbols. Then as long as the system provides a way of decoding that information, you could effectively increase the capacity of any storage mechanism by encoding its contents using said system (analogous to converting every file on a hard disk using some compression algorithm such as LZMA).
But, there's still more to it than that. It goes without saying that any system of compression like what I just described comprises its own information, and therefore needs to be stored somewhere itself.
Since the universe is "all there is" (?), a system of encoding the information contained within it would have to be a part of that very information. This is where I think I hit a mental wall. On the one hand, it seems that you could extract a seemingly unlimited amount of information from finite data—by using a system to encode that data, another system to encode the encoded data, and so on and so forth—whereas on the other, intuition tells me that there must come a point where, if the data as well as the system must share a space, there is no longer any room for either more data or another system of encoding it. The available space becomes too "crowded," so to speak.
Is there a mathematical principle or theorem that answers this question? Is the problem I'm describing (determining a limit on the capacity of material data to store information) defined, analyzed, and/or illuminated by any particular concept(s) in mathematics?
 A: Perhaps you are looking for the holographic principle.  It is conjectural, even at a physical level of rigor, but it puts a bound on the amount of information contained in a region in terms of the region's surface area.
A: The holographic principle has been mentioned so I'll just add a little more info from the physics perspective. 
The Bekenstein entropy bound implies that there is a finite number of information (entropy) in a finite volume of space with finite energy. 
Speaking in terms of entropy, one can see for example that there must be a limit on how many fundamental particles there are. The reasoning is that given a particle made up of sub particles, the total number of degrees of freedom of the particle is the product of the degrees of freedom of each sub particle (are there sub particles with only one degree of freedom?). Since the total number of degrees of freedom must be finite, this implies that one cannot subdivide particles forever. 
There are some particularly striking consequences of these entropy bounds. For example, Verlinde argues that the force of gravity between particles is a result of the holographic principle. This can be thought of as (indirect) physical evidence of the holographic principle at work. 
