Thm : \Sigma is Maximum Satisfiability 
             iff 
      \Sigma is satisfiable and complete.
Proof of above theorem. 

Claim: \Sigma \entails \alpha Iff \Sigma \cup \{ \neg \alpha\} is Not satisfiable. 


Finite models theorem: (FMT)
A set \Sigma \entails some wff \alpha 
implies
there is a finite subset of \Sigma which \entails \alpha.

Proof of FMT using claim above. 

Claim : \Sigma is maximally satisfiable  Iff 
        \Sigma has a unique valuation. 
Left as an exercise.