It solved, in one brilliant flash, the so-called decision problem of the great nineteenth-century German mathematician David Hilbert: Devise a mechanical procedure that could determine whether or not any given mathematical statement was true. When Godel announced his result, its impact on logic and mathematics was sensational and profound. That is, there are statements within arithmetic that cannot be proved true or false within the system arithmetic is undecidable. Godel showed that if a system as rich as arithmetic is consistent, it cannot be complete. Arithmetic furnishes some examples: the axiom of arithmetic that makes, say, 2 + 0 equal to 2 (and likewise, for the addition of zero to any natural number) and the logical rule that enables one to infer, from the proposition if p then q and the proposition p, that q. Such systems are made up of some finite number of statements called axioms, which are accepted as true and 'self-evident,' as well as some equally unobjectionable rules for deriving one statement from another, called rules of inference. In 1931 the Czech logician, then a young professor at the University of Vienna, published a scholarly paper summarizing his doctoral investigations into the properties of formal logical systems.

It is easy to understand why one might come to think that Godel's result would have implications for science. That interpretation of Godel's fundamental contribution to logic seems to be a commonly held belief, but it is simply not so.

He replied that such limits had been established by Kurt Godel. In the early 1990s I mentioned to a senior European scientist that I was interested in understanding the intrinsic limits to scientific knowledge. Barrow Oxford University Press, 1998 279 pages $25.00

IMPOSSIBILITY: THE LIMITS OF SCIENCE AND THE SCIENCE OF LIMITS by John D.

Impossibility by John D. Barrow