"Generalization" of Gödel's Theorem One question came to my mind while arguing with a friend about the necessity of Judges in society (I will explain...)
In the reasoning, we came across the following argument:
"If we put up a good Law System, there should be no need for judges, only an algorithmical processing of the laws."
So I thought:
What if a given Law System could be PROVED not to be both consistent and complete, meaning the following could never hold at the same time:
1- Every possible situation has a definite way to be judged (according to the laws).
2- There is no contradiction in the laws.
This way, the need for judges would be justified.
Of course, as in Gödel's Theorem, we should expect that a "minimum" of laws should be attained, at least.
With all those considerations, my questions are:

0- Does my question even make sense?
1- Are my interpratations of "consistent" and "complete" coherent, in the sense I explained?
2- Is there an answer to this question?
2- What would be the "minimum" of laws?

 A: I assume that you are speaking of an analogy with Godel's Completeness Theorem; if so, you do not need necessarily Arithmetic.
In order to use your analogy, you must (I think) :
1) Axiomatize it, i.e. state a certain amount of Law Principles that you accept as postulates
2) Assume some set of Rules of Proof (for example : deduction rules of First Order Logic)
3) ... and I think is the most difficult part ... regiment your language in order to be able to make precise statements.
Only with this three conditions fulfilled you can try to :
A) check if your Law System is consistent, i.e. try to prove that for NO sentence expressible into "law language" $L$ , it is possible to find a proof (a correct one, according to the assumed Rules of Proof) , starting from the postulates of your Law System , of BOTH $L$ and $\lnot L$ (not-L).
B) regarding completeness, starting again from the assumption that the "law language" is capable of describing in a precise way all possible situations, your intent is to prove that there exist "right" decisions that you cannot reach form the postulates by means only of the Rules of Proof (the algorithmical processing). 
To do this, you must be able to define in some way the class of "right" decisions. 
In a more "formal" way, [rif. S.C.Kleene, Introduction to Metamathematics (1952), pag.131] :

suppose that some property has been defined for the formulas of the system [...]. The system is consistent with respect to the property (or interpretation), if only formulas which have the property are provable. The system is complete with respect to the property , if all formulas which have the property are provable.

