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For a more exact discussion of the notion of translation, cf.Wang [14].
I am indebted to ProfessorChurch for pointing out this known fact to me for the first time. SeeZermelo [16],Grelling [6], andBernays [3].
SeeAckermann [1]. His system does not include M 2, M 4, or M 5. But it presents no special difficulty if we add these axioms (in connection with M 5, cf.Péter [1]).
This assertion becomes obvious, if we reflect that all true statements containing no variables are provable in number theory and 1.22 gives us all the true statements of number theory, as determined by the truth definition inHilbert-Bernays (cf. [8], p. 333). With regard to the rule of infinite induction, ProfessorBernays observes the following two different ways of construing it. On the one hand, the rule may be construed as a part of, so to speak, a metamathematical definition of the system of “true” number-theoretic statements (in classical sense), expressing a property of closure. On the other hand the rule may also be construed as one for derivation, which differs from the usual rules of derivation in that it requires admission of proof schemata as generating proofs. It is by the first interpretation of infinite induction that number theory becomes complete and ω-complete by the addition of 1.22. If we had adopted the second interpretation, it could only be said that there is, after addition of the infinite induction, nodemonstrable ω-incompleteness because any general argument for proving, say, thatH (0),H (1),H (2), ... are all provable would also yield, by infinite induction, a proof for “for allm, H (m)”.
Bernays suggests the following more direct proof of theorem 6. IfP (x) is any arithmetic predicate, then a numbert of a triad can be effectively determined such that for every numberm, we havem η*t if and only ifP (E (m)). Therefore, if η* were recursive, we could effectively decide for every numberk (for which indeedm can be effectively determined so thatk=E (m)) whetherP (k) holds. Thus every arithmetic predicate would be recursive, which by Kleene's well-known results is not the case. I am keeping my original proof because it might be considered more elementary in so far as it does not depend on Kleene's result.
We omit the details of the definition ofK and the proofs of the cases forK (n, 0),K (n, 1), etc., on the following grounds: (1) details of similar definitions and proofs are treated inWang [15] in the general case; (2) the case ofR which we shall consider soon is again similar, although somewhat more complex.
SeeRobert McNaughton, On establishing the consistency of systems, Dissertation, Harvard University, 1951. He observes that in his proof he makes use of certain ideas and methods introduced byI. L. Novak, Fundamenta mathematica,37, 87–110 (1950).
We note that similar considerations also apply toG*. For example, ifp is a given undecidable statement, then the statement “there exists some sety such that for allx,x εy if and only ifp is false andx=x” is also undecidable. So far as we know, the general method of generating equivalent undecidable statements in the above manner was first introduced in a different connection byW. V. Quine (see Journal of symbolic logic,6, 140).
This kind of model is discussed byL. Henkin, Journal of symbolic logic15, 81–91 (1950). In his terminology, a model with natural numbers which are non-regular would be a “non-standard model”. However, we find it difficult to obtain an exact definition of the general notion of standard models.
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Wang, H. Between number theory and set theory. Math. Ann. 126, 385–409 (1953). https://doi.org/10.1007/BF01343173
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DOI: https://doi.org/10.1007/BF01343173