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Shannon and W. Weaver, The Mathematical Theory of Communication. : University of Illinois Press, 1949. 16] M. Blum, \A machine-independent theory of the complexity of recursive functions," J. ACM, vol. 14, pp. 322{336, April 1967. 17] M. A. Arbib and M. Blum, \Machine dependence of degrees of di culty," Proc. AMS, vol. 16, pp. 442{447, June 1965. 18] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Oxford: Oxford University Press, 1962. The following references have come to the author's attention since this lecture was given.

4] |, \Applications of recursive function theory to number theory," Proc. Symp. in Pure Mathematics, vol. 5. : AMS, 1962, pp. 135{138. 5] M. Minsky, Computation: Finite and In nite Machines. : Prentice-Hall, 1967. 6] H. , Theory of Recursive Functions and E ective Computability. New York: McGraw-Hill, 1967. 7] M. A. Arbib, Theories of Abstract Automata. : Prentice-Hall (to be published). 8] R. J. Solomono , \A formal theory of inductive inference," Inform. and Control, vol. 7, pp. 1{22, March 1964 pp.

1 (July 1988), pp. 80{85 by Gregory J. Chaitin Gregory J. Chaitin is on the sta of the IBM Thomas J. Y. He is the principal architect of algorithmic information theory and has just published two books in which the theory's concepts are applied to elucidate the nature of randomness and the limitations of mathematics. This is Chaitin's second article for Scientific American. Abstract It is impossible to prove whether each member of a family of algebraic equations has a nite or an in nite number of solutions: the answers vary randomly and therefore elude mathematical reasoning.

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