Minds, Machines and Gödel1

^{page 112 note 2} See Turing, A. M.: “Computing Machinery and Intelligence”: Mind, 1950, PP. 433–60, reprinted in The World of Mathematics, edited by James R. Newman, pp. 2099–123; and Popper, K. R.: “Indeterminism in Quantum Physics and Classical Physics”; British Journal for Philosophy of Science, Vol.1 (1951), pp.179–88. The question is touched upon by Rosenbloom, Paul; Elements of Mathematical Logic; pp.207–8; Nagel, Ernest and Newman, James R.; Gödel's proof, pp. 100–2; and by Rogers, Hartley; Theory of Recursive Functions and Effective Computability (mimeographed), 1957, Vol.1, pp. 152 ff.

^{page 117 note 1} Gödel's original proof applies; v. § i init. § 6 init. of his Lectures at the Institute of Advanced Study, Princeton, NJ., U.S.A., 1934.

^{page 117 note 2} Mind, 1950, pp. 444–5; Newman, p. 2110.

^{page 118 note 1} For a similar type of argument, see Lucas, J. R.: “The Lesbian Rule”; PHILOSOPHY, 07 1955, pp. 202–6; and “On not worshipping. Facts”; The Philosophical Quarterly, April 1958, p. 144.

^{page 119 note 1} In private conversation.

^{page 119 note 2} Theory of Recursive Functions and Effective Computability, 1957, Vol. I, pp. 152 ff.

^{page 120 note 1} Godel's original proof applies if the rule is such as to generate a primitive recursive class of additional formulae; v. § I init. and § 6 init. of his Lectures at the Institute of Advanced Study, Princeton, N.J., U.S.A., 1934. It is in fact sufficient that the class be recursively enumerable, v. Rosser, Barkley: “Extensions of some theorems of Godel and Church”, Journal of Symbolic Logic, Vol. 1, 1936, pp. 87–91.

^{page 120 note 2} Op. cit., p. 154.

^{page 120 note 3} University of Princeton, N.J., U.S.A. in private conversation.

^{page 121 note 1} See, e.g., Church, Alonzo: Introduction to Mathematical Logic, Princeton. Vol. I, § 17, p. 108.

^{page 125 note 1} Mind, 1950, p. 454; Newman, p. 2117–18.

¹ A paper read to the Oxford Philosophical Society on October 30, 1959.