Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-08T02:04:05.542Z Has data issue: false hasContentIssue false
  • English
  • Français

On external Scott algebras in nonstandard models of Peano arithmetic

Published online by Cambridge University Press:  12 March 2014

Vladimir Kanovei
Vladimir Kanovei*
Affiliation:
Department of Mathematics (Vychislitelnaya), Moscow Transport Engineering Institute, Obraztsova 15, Moscow 101475, Russia, E-mail: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that a necessary and sufficient condition for a countable set of sets of integers to be equal to the algebra of all sets of integers definable in a nonstandard elementary extension of ω by a formula of the PA language which may include the standardness predicate but does not contain nonstandard parameters, is as follows: is closed under arithmetical definability and contains 0(ω) the set of all (Gödel numbers of) true arithmetical sentences.

Some results related to definability of sets of integers in elementary extensions of ω are included.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 2 , June 1996 , pp. 586 - 607
Copyright
Copyright © Association for Symbolic Logic 1996

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]Harrington, L., The constructible reals can be anything, a preprint dated May 1974 with several addendums dated up to 10 1975.Google Scholar
[2]Hinman, P., Recursion theoretic hierarchies, Springer-Verlag, Berlin, 1978.Google Scholar
[3]Kanovei, V., The set of all analytically definable sets of natural numbers can be defined analytically, Mathematics of the USSR-Izvestiya, vol. 15 (1980), pp. 469–500.CrossRefGoogle Scholar
[4]Kaye, R. W., Models of Peano arithmetic, Oxford logic guides, vol. 15, Oxford Science Publications, 1991.Google Scholar
[5]Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory, Proceedings of the Fifth Symposium in Pure Mathematics of the American Mathematical Society, 1962, pp. 117–121.Google Scholar