Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T13:01:46.757Z Has data issue: false hasContentIssue false
  • English
  • Français

AUTOMORPHISM GROUPS OF SATURATED MODELS OF PEANO ARITHMETIC

Published online by Cambridge University Press:  25 June 2014

ERMEK S. NURKHAIDAROV  and
JAMES H. SCHMERL
ERMEK S. NURKHAIDAROV
Affiliation:
PENNSYLVANIA STATE UNIVERSITY, MONT ALTO, 1 CAMPUS DR., MONT ALTO, PA 17237E-mail: esn1@psu.edu
JAMES H. SCHMERL
Affiliation:
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CONNECTICUT STORRS, CT 06269-3009E-mail: james.schmerl@uconn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Let κ be the cardinality of some saturated model of Peano Arithmetic. There is a set of ${2^{{\aleph _0}}}$ saturated models of PA, each having cardinality κ, such that whenever M and N are two distinct models from this set, then Aut(${\cal M}$) ≇ Aut ($${\cal N}$$).

Keywords

Peano arithmeticsaturated modelsautomorphism groups
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 2 , June 2014 , pp. 561 - 584
Copyright
Copyright © The Association for Symbolic Logic 2014 

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

Bamber, Nicholas and Kotlarski, Henryk, On interstices of countable arithmetically saturated models of Peano arithmetic. Mathematical Logic Quarterly, vol. 43 (1997), pp. 525–540.Google Scholar
Kaye, Richard, A Galois correspondence for countable recursively saturated models of Peano arithmetic, Automorphisms of first-order structures (Kaye, Richard and Macpherson, Dugald, editors), Oxford University Press, New York, 1994, pp. 293–312.Google Scholar
Kaye, Richard and Macpherson, Dugald, Automorphisms of first-order structures, Oxford University Press, New York, 1994.CrossRefGoogle Scholar
Kossak, Roman and Schmerl, James H., The automorphism group of an arithmetically saturated model of Peano arithmetic. Journal of the London Mathematical Society, vol. 52 (1995), no. 2, pp. 235–244.CrossRefGoogle Scholar
Kossak, Roman and Schmerl, James H., Arithmetically saturated models of arithmetic. Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 531–546.Google Scholar
Kossak, Roman and Schmerl, James H., The structure of models of Peano arithmetic, Oxford University Press, Oxford, 2006.Google Scholar
Kotlarski, Henryk and Kaye, Richard, Automorphisms of models of true arithmetic: Recognizing some basic open subgroups. Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 1–14.Google Scholar
Lascar, Daniel, The small index property and recursively saturated models of Peano arithmetic, Automorphisms of first-order structures (Kaye, Richard and Macpherson, Dugald, editors), Oxford University Press, New York, 1994, pp. 281–292.Google Scholar
Lascar, Daniel, The group of automorphisms of a relational saturated structure, Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Series C Mathematical Physical Sciences, vol. 411, Kluwer Academic Publisher, Dordrecht, 1993, pp. 225–236.Google Scholar
Lascar, Daniel and Shelah, Saharon, Uncountable saturated structures have the small index property. Bulletin of the London Mathematical Society, vol. 25 (1993), pp. 125–131.CrossRefGoogle Scholar
Melles, Garvin and Shelah, Saharon, A saturated model of an unsuperstable theory of cardinality greater than its theory has the small index property. Proceedings of the London Mathematical Society, vol. 69 (1994), no. 3, pp. 449–463.CrossRefGoogle Scholar
Nurkhaidarov, Ermek S., Automorphism groups of arithmetically saturated models, this Journal, vol. 71 (2006), pp. 203–216.Google Scholar
Nurkhaidarov, Ermek S. and Shochat, Erez, Automorphisms of saturated and boundedly saturated models of arithmetic. Notre Dame Journal of Formal Logic, vol. 52 (2011), pp. 315–329.CrossRefGoogle Scholar
Nurkhaidarov, Ermek S. and Shochat, Erez, Closed normal subgroups of the automorphism group of a saturated model of Peano Arithmetic. Notre Dame Journal of Formal Logic (forthcoming).Google Scholar
Odifreddi, P. G., Classical recursion theory, vol. II, Studies in Logic and the Foundations of Mathematics, vol. 143, North-Holland Publishing Company, Amsterdam, 1999.Google Scholar
Scott, Dana, Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory (Dekker, J. C. E., editor), Proceeding of symposia in pure mathematics, vol. V, American Mathematical Society, Providence, 1962, pp. 117–122.Google Scholar
Slaman, T. A. and Woodin, W. H., Definability in degree structures, unpublished manuscript.Google Scholar
Smoryński, Craig, Back-and-forth inside a recursively saturated model of arithmetic, Logic colloquium ’80 (Dalen, D. van, editor), North-Holland, Amsterdam, 1982, pp. 273–278.Google Scholar