Regarding inexpressibility, I have come across the term functionally complete, or Turing complete. It seems to be the ability to express any function.
Now, in my first year of undergrad, we covered Boolean algebra (in engineering math, so it's probably different from the Boolean algebras I keep hearing about). Boolean algebra was defined for us to be turning complete. I remember there was a proof, but I don't remember the proof.
For us, it was defined as everything you can do with $A,B,C,\dots$ $\bar X$ (negation), $\cdot$ (AND), and $+$ (OR). There were no axioms, and the rules were not …