[felixhandte]

Sure, but Gödel encoding is pretty much purely a theoretical exercise. I'm not sure anyone anywhere has ever practically manipulated Gödel-encoded expressions in a useful way. His original scheme also has the problem that prime factorization is rather computationally challenging--it is after all the basis of the RSA cryptosystem.

Whereas Arithmetic encoding is actually practical, extensively used, and a direct analogue to the stick.

[sdfsdfsfds]

I'm aware of arithmetic encoding and it is definitely the most compelling example of encoding arbitrary data with a single number. On the other hand, there is a lot more to arithmetic coding than the ability to encode lists of numbers—all the considerations involving the context of each symbol, which are essential to the process of compression. I just felt that it might be helpful to give an example which didn't implicate all that complex compression apparatus.