16 The addition of the Peano axioms, like all the other changes made in the system PM, serves only to simplify the proof and can in principle be dispensed with.

17 It is presupposed that for every variable type denumerably many signs are available.

18 Unhomogeneous relations could also be defined in this manner, e.g. a relation between individuals and classes as a class of elements of the form: ((x 2 ),((x 1 ),x 2 )). As a simple consideration shows, all the provable propositions about relations in PM are also provable in this fashion.

18a Thus x " (a) is also a formula if x does not occur, or does not occur free, in a. In that case x " (a) naturally means the same as a.

19 With regard to this definition (and others like it occurring later), cf. J. Lukasiewicz and A. Tarski, 'Untersuchungen über den Aussagenkalkül', Comptes Rendus des séances de la Soeiété des Sciences et des Lettres de Varsovie XXIII, 1930, Cl. 111.

20 Where v does not occur in a as a free variable, we must put Subst a(v|b) = a. Note that "Subst" is a sign belonging to metamathematics.

21 As in PM I, *13, x 1 = y 1 is to be thought of as defined by x 2 " (x 2 (x 1 ) É x 2 (y 1 )) (and similarly for higher types.)

22 To obtain the axioms from the schemata presented (and in the cases of II, III and IV, after carrying out the permitted substitutions), one must therefore still



1. eliminate the abbreviations

2. add the suppressqd brackets.



23 c is therefore either a variable or 0 or a sign of the form ¦ ¦ u where u is either 0 or a variable of type 1. With regard to the concept "free (bound) at a place in a" cf. section I A5 of the work cited in footnote 24.

24 The rule of substitution becomes superfluous, since we have already dealt with all possible substitutions in the axioms themselves (as is also done in J. v. Neumann, 'Zur Hilbertschen Beweistheorie', Math. Zeitschr. 26, 1927).

25 I.e. its field of definition is the class of non-negative whole numbers (or n-tuples of such), respectively, and its values are non-negative whole numbers.

26 In what follows, small italic letters (with or without indices) are always variables for non-negative whole numbers (failing an express statement to the contrary). [Italics omitted.]

27 More precisely, by substitution of certain of the foregoing functions in the empty places of the preceding, e.g. f k (x 1 ,x 2 ) = f p [ f q (x 1 ,x 1 ), f r (x 2 )] (p, q, r  k). Not all the variables on the left-hand side must also occur on the right (and similarly in the recursion schema (2)).

28 We include classes among relations (one-place relations). Recursive relations R naturally have the property that for every specific n-tuple of numbers it can be decided whether R(x 1 x n ) holds or not.

29 For all considerations as to content (more especially also of a metamathematical kind) the Hilbertian symbolism is used, cf. Hilbert-Ackermann, Grundzüge der theoretischen Logik, Berlin 1928.

30 We use [greek] letters c , h , as abbreviations for given n-tuple sets of variables, e.g. x 1 , x 2 x n .

31 We take it to be recognized that the functions x+y (addition) and x.y (multiplication) are recursive.

32 a cannot take values other than 0 and 1, as is evident from the definition of a .

33 The sign º is used to mean "equivalence by definition", and therefore does duty in definitions either for = or for ~ [not the negation symbol] (otherwise the symbolism is Hilbertian).

34 Wherever in the following definitions one of the signs (x), ( $ x), e x occurs, it is followed by a limitation on the value of x. This limitation merely serves to ensure the recursive nature of the concept defined. (Cf. Proposition IV.) On the other hand, the range of the defined concept would almost always remain unaffected by its omission.

34a For 0 < n £ z, where z is the number of distinct prime numbers dividing into x. Note that for n = z+1 , n Pr x = 0.

34b m,n £ x stands for: m £ x & n £ x (and similarly for more than two variables).

35 The limitation n £ (Pr[l(x)]2x.[l(x)]2 means roughly this: The length of the shortest series of formulae belonging to x can at most be equal to the number of constituent formulae of x. There are however at most l(x) constituent formulae of length 1, at most l(x)-1 of length 2, etc. and in all, therefore, at most 1 ¤ 2 [l(x){l(x)+1}] £ [l(x)]2. The prime numbers in n can therefore all be assumed smaller that Pr{[l(x)]2}, their number £ [l(x)]2 and their exponents (which are constituent formulae of x) £ x.

36 Where v is not a variable or x not a formula, then Sb(x v|y) = x.

37 Instead of Sb[Sb[x v|y] z|y] we write: Sb(x v|y w|z) (and similarly for more than two variables).

38 The variables u 1 u n could be arbitrarily allotted. There is always, e.g., an r with the free variables 17, 19, 23 etc., for which (3) and (4) hold.

39 Proposition V naturally is based on the fact that for any recursive relation R, it is decidable, for every n-tuple of numbers, from the axioms of the system P, whether the relation R holds or not.

40 From this there follows immediately its validity for every recursive relation, since any such relation is equivalent to 0 = f (x 1 x n ), where f is recursive.

41 In the precise development of this proof, r is naturally defined, not by the roundabout route of indicating its content, but by its purely formal constitution.

42 Which thus, as regards content, expresses the existence of this relation.

43 r is derived in fact, from the recursive relation-sign q on replacement of a variable by a determinate number (p).

44 The operations Gen and Sb are naturally always commutative, wherever they refer to different variables.

45 "x is c-provable" signifies: x Î Flg(c), which, by (7), states the same as Bew c (x).

45a Since all existential assertions occurring in the proof are based on Proposition V, which, as can easily be seen, is intuitionistically unobjectionable.

46 Thus the existence of consistent and not w -consistent c's can naturally be proved only on the assumption that, in general, consistent c's do exist (i.e. that P is consistent).

47 The proof of assumption 1 is here even simpler than that for the system P, since there is only one kind of basic variable (or two for J. v. Neumann).

48 Cf. Problem III in D. Hilbert's lecture: 'Probleme der Grundlegung der Mathematik', Math. Ann. 102.