LOGIC in 1926 showed that the propositional system of Principia Mathematica was consistent, and that four of its axioms were independent of each other, but the fifth was deducible as a thesis from the remaining four The method of proving consistency and independence depends upon treating the axioms and theorems of a deductive system simply as abstract formulae, and treating the rules of the system simply as mechanical procedures for obtaining one formula from another The properties of the system are then explored by offering a set of objects as a model, or interpretation, of the abstract calculus The elements of the system are mapped on to the objects and their relations in such a way as to satisfy, or bring out true, the formulae of the system A formula P will entail a formula Q if and only if all interpretations that satisfy P also satisfy Q This model-theoretic approach to logic gradually assumed an importance equal to that of the earlier approach that had focused on the notion of proof A third property of deductive systems that was explored by logicians in the inter-war years was that of completeness An axiomatic presentation of the propositional calculus is complete if and only if every truth-table tautology is provable within the system Hilbert and Ackermann in 1928 offered a proof that the propositional calculus of Principia Mathematica was in this sense complete Indeed, it was complete also in the stronger sense that if we add any nontautologous formula as an axiom, we reach a contradiction In 1930 Kurt Goădel proved that first-order predicate calculus, the logic of quantification, was complete in the weaker, but not the stronger, sense The question now arose: was arithmetic, like general logic, a complete system? Frege, Russell, and Whitehead had hoped that they had established that arithmetic was a branch of logic Russell wrote, ‘If there are still those who not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins’ (IMP 194–5) If arithmetic was a branch of logic, and if logic was complete, then arithmetic should be a complete system too Goădel, in an epoch-making paper of 1931, showed that it was not, and could not be turned into one By an ingenious device he constructed a formula within the system of Principia that can be shown to be true and yet is not provable within the system: a formula that in effect says of itself that it is unprovable He did this by showing how to turn formulae of the logical system into statements of arithmetic by associating the signs of Principia with 115