[throwway120385]

Are you sure the number of possible expressions is finite? Or is it countably infinite? I could in theory make an arbitrarily long expression out of the four fundamental operators, and because I can derive -1 from a finite set of operations it is possible to derive any integer from any even integer. You can also derive zero from the rules.

[lmm]

> I could in theory make an arbitrarily long expression out of the four fundamental operators

Not with only four inputs you can't. You can only have three operations, because you have no way of getting another input parameter.