Consistency and Completeness of W
It is easily established that every thesis of W is valid. (Note that wffs containing ‘0’ present no problem: in testing them for validity each occurrence of ‘0’ is always assigned the value false.) The consistency of W with respect to negation then follows by the definition of negation and the consideration that for any wff, X,at most one of X and X ⊃ 0 can be valid. It then follows that W is also consistent absolutely and in the sense of Post.