Consistency and Completeness of W
GEORGE E. HUGHES, D. G. Londey · 2019
To show that every thesis of Principia Mathematica (PM) is a thesis of W, the authors notice that the theses of PM are the axioms of that system together with their transforms under Substitution and Detachment. Then since the Transformation Rules of the two systems are the same, they know that if the axioms of PM are theses of W, so are all their transforms under those rules. As regards completeness, it is possible to show directly that every valid well-formed formula of W is a thesis of W. However, the authors shall show this indirectly by showing that every thesis of PM is a thesis of W – i.e. that W contains PM. This result provides a basis for proving that W is weakly complete. The proof that this rule holds for W is very similar in structure to the earlier proof that it holds for PM.