By Ernest Nagel, Patrick Suppes and Alfred Tarski (Eds.)

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Completely defined. By the hypo indo applied for each O'i-2, Supposition (II) is met; and hence, by the result of the first induction, 1'2(Y,(lXi(Pi-I)h~I' d, V, V) for pi-! ] ... ) is the Godel number of the next situation; and hence, by the hypo indo applied to the whole expression in the case, im i will continue to a terminal situation, etc. 5. The function A1' i - 3 • . substituted in Case f+2 (f = 3, ... , r) of (2) is of the form A1' i - 3 • • • for S4'·(f-2) (f-2 = 1, ... 8, and the substitutions of AO'i-2 .

2. 34 MATHEMATICAL LOGIC (III) Unlike the situation with the reduction principle, there is no obvious relationship between the first separation principles of pure logic and of recursive function theory. But the knowledge of how the separation properties behave in recursive function theory can at least suggest how they might behave in pure logic. Illustrative of this is the case of SePr(l\D. Since Sep, (lID is known to be false, it is suggested that Sep, (I\~) should also be false. A study of this question leads to confirmation in two different ways.

2. 34 MATHEMATICAL LOGIC (III) Unlike the situation with the reduction principle, there is no obvious relationship between the first separation principles of pure logic and of recursive function theory. But the knowledge of how the separation properties behave in recursive function theory can at least suggest how they might behave in pure logic. Illustrative of this is the case of SePr(l\D. Since Sep, (lID is known to be false, it is suggested that Sep, (I\~) should also be false. A study of this question leads to confirmation in two different ways.