By C.A. Reiter, W.R. Jones

This booklet will be of curiosity to arithmetic scientists operating within the parts of linear algebra, summary algebra, quantity conception, numerical research, operations study and mathematical modelling

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**Example text**

A) The function A grows faster than any one-argument primitive recursive function, hence it cannot be primitive recursive. A b) Some small values of have bsen obtained (for example, i ( 1 ) = d.. "3, where power 2 appears 7 3, A(2) = 7, A(3) = 2'&3 = 61, 2 times). 0. c) Each A,,+1 grows more rapidly than A,,, since the former cannot be reached from the latter without the aid of itself. Nevertheless, all A,, are primitive recursive in contrast to A which cannot be primitive recursive. Returning to the PARIS-HARRINGTON example we can see that the functions R i and 2 have roughly the same rate of growth.

The codification scheme J is often called the CANTOR numbering and the number J(z,y) is referred to a8 the CANTOR number associated to the pair ( 2 , ~ ) . 34 Remark. Caludc The primitive recursive function "1 is also a pairing function. 8) Proposition. If r : p + N is a pairing function, then from the equalities u l ( z ) = u l ( z ' )and u 2 ( z ) = c2(z')we infer z = 2'. Proof. We have: z = 7r(u1(2),u~(2)) = a(ul(z'),u2(i)) = 2'. 0 Proposition. Let a : N --c N be a pairing function and E {ul,u2}.

In the opposite case we use again the auxiliary function f Let k be the smallest natural number for which g k ( a )= 0 , and deflne f by cases as follows: . (z+l) = g ( f - ( z ) ) . s -- a 5 5 G, * * * is The increasing sequence of classes G I G1 'z called the ACKERMANN-PETER hierarchy. The main results concerning the classes G , are due t o GEORGIEVA [1978a]. PAS) Theorem. (PETER [1957)) The diagonal function A ( z ) = A ( z , z ) is not primitive recursive. A:N - RV, 21 Chapter 1 Proof. h u m e , for the sake of contradiction, that 2 ia primitive recuris in G,.