No CrossRef data available.
Article contents
SELF-EMBEDDINGS OF MODELS OF ARITHMETIC; FIXED POINTS, SMALL SUBMODELS, AND EXTENDABILITY
Published online by Cambridge University Press: 22 December 2022
Abstract
In this paper we will show that for every cut I of any countable nonstandard model
$\mathcal {M}$
of
$\mathrm {I}\Sigma _{1}$
, each I-small
$\Sigma _{1}$
-elementary submodel of
$\mathcal {M}$
is of the form of the set of fixed points of some proper initial self-embedding of
$\mathcal {M}$
iff I is a strong cut of
$\mathcal {M}$
. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model
$\mathcal {M}$
of
$ \mathrm {I}\Sigma _{1} $
. In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of
$ \mathrm {I}\Sigma _{1} $
to larger models.
- Type
- Article
- Information
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
References
Bahrami, S. and Enayat, A.,
Fixed points of self-embeddings of models of arithmetic
.
Annals of Pure and Applied Logic
, vol. 169 (2018), pp. 487–513.CrossRefGoogle Scholar
Barwise, J. and Schlipf, J., On recursively saturated models of arithmetic,
Model Theory and Algebra: A Memorial Tribute to A. Robinson
(Saracino, D. and Weispfenning, V., editors), Lecture Notes in Mathematics, vol. 498, Springer, Berlin, 1976, pp. 42–55.CrossRefGoogle Scholar
Dimitracopoulos, C. and Paris, J.,
A note on a theorem of H. Friedman
.
Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
, vol. 34 (1988), no. 1, pp. 13–17.CrossRefGoogle Scholar
Enayat, A.,
Automorphisms of models of arithmetic: A unified view
.
Annals of Pure and Applied Logic
, vol. 145 (2007), pp. 16–36.CrossRefGoogle Scholar
Friedman, H.,
Countable models of set theories
,
Cambridge Summer School in Mathematical Logic
(Mathias, A. R. D. and Rogers, H., editors), Lecture Notes in Mathematics, vol. 337, Springer, Berlin, 1973, pp. 539–573.CrossRefGoogle Scholar
Hájek, P. and Pudlák, P.,
Metamathematics of First Order Arithmetic
, Springer, Heidelberg, 1993.CrossRefGoogle Scholar
Kaye, R., Kossak, R., and Kotlarski, H.,
Automorphisms of recursively saturated models of arithmetic
.
Annals of Pure and Applied Logic
, vol. 55 (1991), pp. 67–99.CrossRefGoogle Scholar
Kirby, L. and Paris, J.,
${\varSigma}_n$
-collection schemas in arithmetic
,
Logic Colloquium ’77
(A. Macintyre, L. Pacholski, and J. Paris, editors), North-Holland, Amsterdam, 1978, pp. 199–209.Google Scholar
${\varSigma}_n$
-collection schemas in arithmetic
,
Logic Colloquium ’77
(A. Macintyre, L. Pacholski, and J. Paris, editors), North-Holland, Amsterdam, 1978, pp. 199–209.Google ScholarKossak, R. and Kotlarski, H., Results on automorphisms of recursively saturated models of
$PA$
.
Fundamenta Mathematicae
, vol. 129 (1988), no. 1, pp. 9–15.CrossRefGoogle Scholar
$PA$
.
Fundamenta Mathematicae
, vol. 129 (1988), no. 1, pp. 9–15.CrossRefGoogle ScholarKossak, R. and Kotlarski, H.,
On extending automorphisms of models of Peano arithmetic
.
Fundamenta Mathematicae
, vol. 149 (1996), no. 3, pp. 245–263.CrossRefGoogle Scholar
Kossak, R. and Schmerl, J.,
Arithmetically saturated models of arithmetic
,
Notre Dame Journal of Formal Logic
, vol. 36 (1995), no. 4, pp. 531–546.CrossRefGoogle Scholar
Kossak, R. and Schmerl, J.,
The Structure of Models of Peano Arithmetic
, Oxford Logic Guides, vol. 50, Clarendon Press, Oxford, 2006.CrossRefGoogle Scholar
Lascar, D.,
The small index property and recursively saturated models of Peano arithmetic
,
Automorphisms of First-Order Structures
(Kaye, R. and Macpherson, D., editors), Oxford University Press, New York, 1994, pp. 281–292.Google Scholar
Smoryński, C., Recursively saturated nonstandard models of arithmetic, this Journal, 46 (1981), no. 2, pp. 259–286.Google Scholar
Wilkie, A.,
On the theories of end-extensions of models of arithmetic
,
Set Theory and Hierarchy Theory V
(Lachlan, A., Srebrny, M., and Zarach, A., editors), Lecture Notes in Mathematics, vol. 619, Springer, Berlin, 1977, pp. 305–310.CrossRefGoogle Scholar