Diagonalization and Self-Reference by Raymond M. Smullyan

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By Raymond M. Smullyan

This booklet provides a scientific, unified therapy of fastened issues as they take place in G?del's incompleteness proofs, recursion thought, combinatory good judgment, semantics, and metamathematics. filled with instructive difficulties and strategies, the booklet bargains a superb creation to the topic and highlights contemporary learn.

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2 directly as follows. Given a, take b satisfying (2). Then, taking b for y, 0(b, b) ='(a, d(b)) But also by (1), O(d(b)) = 0(b, b). Therefore O(d(b)) = 0(a, d(b)), so O(c) = 0(a, c) for c = d(b). 1 without any further diagonalization. a.. The solution of Problem 4 is that special case of the above theorem in which N is the set of positive integers, W is the set of subsets of N, q(x) is w, and 0(x, y) is the set of all z such that Rx(z, y). 2 in turn has the following corollary. 3 Let - be an equivalence relation on a set N and F(x, y) be a function from ordered pairs of elements of N to elements of N.

Since for every predicate K there is some expression X such that H°(X) -_ H (K (X)), we can take H° for K, and hence there is some expression X (not necessarily a predicate) such that H°(X) - H(H° (X)). Thus H° (X) is a fixed point of H. An application Let us see how this applies to 1Z-systems. Although HR is not a diagonalizer of H, it is a near diagonalizer of H, since for any predicate K, HR*K* - H*K*K*, and so if we take K* for X, then HR*X = H*K*X, and so H°(X) - H(K(X)) where H° is HR. And so HR is a near diagonalizer of H.

Thus for all numbers x and y, Rb(x, Y) R(x, d(y)). Therefore Rb(x, b) H R(x, d(b)). But also by (1), x E wd(b) H Rb(x, b). Therefore x E wd(b) H R(x, d(b)). And so for n = d(b), wn, = {x : R(x, n)}. Take a number b satisfying (2). Then take b for x and d(b) for y, and then by (2), if Robot d(b) creates Robot F(b, b), then Robot d(b) creates Robot d(b). But Robot d(b) does create Robot F(b, b) by (1), and so Robot d(b) does create Robot d(b). Thus Pd(b) is a program for a self-duplicating Q.. 5 robot.

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