1 - Introduction

Summary

Inexhaustibility: the positive side of Incompleteness

In 1931, Kurt Gödel presented his famous incompleteness theorem, which has since had a great and continuing impact on logic and the philosophy of mathematics, and has also like no other result in formal logic caught the interest and imagination of the general public.

Gödel presented two results in his 1931 paper, usually referred to as the first and the second incompleteness theorem. The proof of the first incompleteness theorem shows that for every consistent formal axiomatic theory in a wide class of such theories, there is at least one statement which can be formulated in the language of the theory but can neither be proved nor disproved in the theory. Such a statement is said to be undecidable in the theory. By a consistent formal theory is meant one in which no logical contradiction — both a statement A and its negation not-A — can be proved. The second theorem states that for a wide class of such theories T, if T is consistent, the consistency of T cannot be proved using only the axioms of the theory T itself.

In discussions of the meaning and implications of these theorems, their negative or limiting aspects are most often kept in the foreground: a formal theory of arithmetic cannot be complete, a theory cannot be proved consistent using only the resources of that theory. But the second incompleteness theorem also has a positive aspect, which was emphasized by Gödel (Collected Works, vol. III, p. 309, italics in the original):

It is this theorem [the second incompleteness theorem] which makes the incompletability of mathematics particularly evident. For, it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If somebody makes such a statement he contradicts himself. For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from his axioms.