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By H. G. Hadley, T. M. Whitin

Mathematical types of stock structures

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M1t is placed on the length of tape or on the time required for computation of a particular value of the function, except that it be finite in each case. Further restrictions imposed on the character of the machine are more or less clearly either non-essential or necessarily contained in the require~ent of finiteness. ability and general recursiveness (attention being confined to functions of one argument for which the range of arguments con- III. »tBDA-DEFINABILITY sists of all positive integers)' is proved in Turing [57].

Finitions in §5). ns of addition, multiplication, and exponentiation are due to Rosser (see Kleene [35]). "lSfonnations. The reader may also verify that, for a:n:y formulas L, ftt, If (whether representing positive integers or not): [L+Nl+/f conv L+l'/tt+/f], [L+/llJ •/f conv [L •IY]+[/tt•/f}, Lffl+lf conv L/tt•L", tftt•/f conv [L lf3/tt, SN conv 1+/It 9. ORDERED PAIRS AND TRIADS, THE PREDECESSOR FUNCTION. b/c/d). §9. onv If. Verification of this depends on the observation that, if is a formula.

D. AB. The foregoing tacitly assumes that A and B do not contain d as a free variable. The modification necessary for the contrary case is, however, obvious. This completes the proof of 12 I. We define the combination belonging to a well-formed :fo:nnula, by recursion as follows: § 1 3. PRIMITIVE SETS OF FORMULAS 1+5 The combination belonging to x is x (where x is any variable). ( 2) The combination belonging to FA is F' A'• where F' and A' are the combinations belonging to F and A respectively.

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