Logic

Mathematical Interpretation of Formal Systems by TH. Skolem

By TH. Skolem

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Y,,) for all m. The reason for this somewhat lengthy definition of "w-consistency" is given on p. 41 of [7]. Incidentally, the proof below gives a solution to problem 2 of JSL 17 (1952), p. 160. An analogous but simpler argument establishes the w-consistency of the elementary quantification theory of addition and multiplication (without induction). We show that this argument cannot be formalized in Z while it is well known that the consistency of this quantification theory can be proved in Z. Since the proof uses the notion of a computable functional it seems desirable to begin with a section on this notion.

We show that this argument cannot be formalized in Z while it is well known that the consistency of this quantification theory can be proved in Z. Since the proof uses the notion of a computable functional it seems desirable to begin with a section on this notion. § 5. Computable Functionals Informal Idea. (i) By a "computable function" with integral arguments and values is meant a method which provides for any integer 0("> an integer O(m,,> (its value). g. ,A(X)] in the system concerned; in other words, representations of integers by ,u-symbol expressions are (properly) ignored in the definition of computability.

59 Enumeration in a Formal System An interesting question is to examine the possibility of enumerating in a system L all the elements of a model of L. This would clearly be a fertile ground for the operation of the diagonal argument. For instance, if the axiom of the system L is just the formula p given in the preceding section, then an enumeration of a model of L would be provided by (say) a function F such that: F(I)=A (A being the empty set), F(2 a)= f[F(a}], F(3 b)=g[F(b)], F(2 a . 5 C)=h[F(a), F(b), F(c)].

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