## Concluding Reflections

The import of Gödel’s conclusions is far-reaching, though it has not yet been fully fathomed. These conclusions show that the prospect of ﬁnding for every deductive system (and, in particular, for a system in which the whole of arithmetic can be expressed) an absolute proof of consistency that satisﬁes the ﬁnitistic requirements of Hilbert’s proposal, though not logically impossible, is most unlikely.1 They show also that there is an endless number of true arithmetical statements which cannot be formally deduced from any given set of axioms by a closed set of rules of inference. It follows that an axiomatic approach to

number theory, for example, cannot exhaust the domain of arithmetical truth. It follows, also, that what we understand by the process of mathematical proof does not coincide with the exploitation of a formalized axiomatic method. A formalized axiomatic procedure is based on an initially determined and ﬁxed set of axioms and transformation rules. As Gödel’s own arguments show, no antecedent limits can be placed on the inventiveness of mathematicians in devising new rules of proof. Consequently, no ﬁnal account can be given of the precise logical form of valid mathematical demonstrations. In the light of these circumstances, whether an all-inclusive deﬁnition of mathematical or logical truth can be devised, and whether, as Gödel himself appears to believe, only a thoroughgoing philosophical “realism” of the ancient Platonic type can supply an adequate deﬁnition, are problems still under debate and too diﬃcult for further consideration here.2