There is a finite amount of information in the Universe

Theory: The amount of information in the universe is finite.

Assumption: There is a finite amount of mass/energy (hereafter called "stuff") in the universe.

Groundwork. First, we must examine information theory, and specifically the concept of entropy. For this analysis, we will accept the definition of entropy to be synonymous with information.

Example. Papa and Momma have a biological son, Joe, 15, and a biological daughter, Elenka, 12. The following is the "information" encoding in this sentence:
• Joe is Papa's and Momma's son.
• Joe is 15 years old.
• Elenka is Papa's and Momma's daughter.
• Elenka is 12 years old.
That's it. There are a much larger (potentially infinite) number of statements we could make; however, none of these have new information.
• Joe is male (This is encoded in Joe being a son).
• Elenka is younger than Joe (This is encoded in their ages).
• Momma is older than 15 (This is encoded in the biological mother relationship).
Got it so far?

Given that there is a finite amount of stuff in the universe, if there are a finite number of states the stuff can exist in, there is a finite amount of information in the universe.

But wait! Space is infinite! The universe is unbounded! Of course. However, there is a finite amount of stuff in the universe. Being finite, it can only occupy a finite amount of space. One piece of information: any space unoccupied by stuff is empty accounts for the remainder of infinite space.

Sure, that sounds like handwaving, but it isn't. The number 1 has an implicit infinite number of zeros preceding it, but they do not encode any information. Neither does empty space.

Okay, now we come to the hard part. If we accept the lemma "there are a finite number of states the stuff can exist in," we're done, but that would be cheating. First, we must accept that stuff is not infinitely divisible. If, in fact, there are basic units of stuff (quarks, photons, tachyons, et al), each of these units has some simple state:

I'm not a physicist, nor will I ever be. However, I will take on good authority the truth of this fact. If so, then,

There is a finite amount of information in the universe

QED.

To respond to Stealth Munchkin:

SM purports that the infinite series of natural numbers, by their mere existence, represent an infinite amount of information. Not so. If one attempted to label each number in a non-trivial fashion, one might generate more information. The fact that 762 is the number after 761 in radix 10 is trivial. There is no information involved.

If, however, one were to attempt to create a piece of information for every number, where would you store it? Every brain, every piece of silicon, every wavicle would be full, and you'd still only have encoded a finite amount of information.