Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-28T07:19:40.549Z Has data issue: false hasContentIssue false
  • English
  • Français

On models of arithmetic—Answers to two problems raised by H. Gaifman

Published online by Cambridge University Press:  12 March 2014

Alex Wilkie
Alex Wilkie*
Affiliation:
University of Leicester, Leicester, England
Get access Rights & Permissions [Opens in a new window]

Extract

In a recent paper [3] H. Gaifman investigated some model theoretic consequences of Matijasevič's theorem [5], and posed some further problems which naturally arise. We provide here partial answers to two of these problems, the results having been previously announced in the postscript of [3].

Firstly, it is shown in [3] that if M1 and M2 are models of the Peano axioms P and M1 ⊆ M2, then M1 is closed under the recursive functions of M2. The converse of this statement is false. Moreover, Gaifman asks: Is every initial segment of a model M of P which is closed under the recursive functions of M (or the ∑n-definable functions) also a model of P? We show that this is false and our method gives, en route, another proof of a theorem of Rabin [7] stating the P is not implied by any consistent set of ∑n sentences for any n.

Secondly, we partially answer a question posed on p. 129 of [3] by proving (some-what more than) every countable nonstandard model of P has an end extension in which a diophantine equation, not solvable in the original model, has a solution. We can, in fact, take the new model to be isomorphic to the original one. This generalises (apart from the countability restriction) a theorem of Rabin [6].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 1 , March 1975 , pp. 41 - 47
Copyright
Copyright © Association for Symbolic Logic 1975

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Friedman, H., Countable models of set theories, Cambridge Summer School in Mathematical Logic, Springer-Verlag Lecture Notes in Mathematics, no. 337, 1973, pp. 539–573.Google Scholar
[2]Gaifman, H., On local arithmetical functions and their applications for constructing types of Peano arithmetic, Mathematical logic and foundations of set theory, North-Holland, Amsterdam, 1968, pp. 105–121.Google Scholar
[3]Gaifman, H., A note on models and submodels of arithmetic, Conference in Mathematical Logic, London, 1970.Google Scholar
[4]Kleene, S. C., Introduction to metamathematics, North-Holland, Amsterdam, 1967.Google Scholar
[5]Matijasevič, Yu. V., Diophantine representation of recursively enumerable predicates, Proceedings of the Second Scandinavian Logic Symposium, North-Holland, Amsterdam, 1971, pp. 171–178.CrossRefGoogle Scholar
[6]Rabin, M. O., Diophantine equations and non-standard models of arithmetic, Logic, methodology and philosophy of science: Proceedings of the 1960 International Congress, Stanford, Calif., 1962, pp. 151–158.Google Scholar
[7]Rabin, M. O., Non-standard models and the independence of the induction axiom, Essays on foundations of mathematics, Jerusalem, 1966, pp. 287–299.Google Scholar
[8]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar