Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T18:51:59.964Z Has data issue: false hasContentIssue false
  • English
  • Français

AUTOMORPHISM GROUPS OF COUNTABLE ARITHMETICALLY SATURATED MODELS OF PEANO ARITHMETIC

Published online by Cambridge University Press:  22 December 2015

JAMES H. SCHMERL
JAMES H. SCHMERL*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CONNECTICUT STORRS, CT 06269, USAE-mail: james.schmerl@uconn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

If ${\cal M},{\cal N}$ are countable, arithmetically saturated models of Peano Arithmetic and ${\rm{Aut}}\left( {\cal M} \right) \cong {\rm{Aut}}\left( {\cal N} \right)$, then the Turing-jumps of ${\rm{Th}}\left( {\cal M} \right)$ and ${\rm{Th}}\left( {\cal N} \right)$ are recursively equivalent.

Keywords

Peano Arithmeticarithmetically saturatedautomorphism group
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1411 - 1434
Copyright
Copyright © The Association for Symbolic Logic 2015 

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

Ash, C. J. and Knight, J., Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland, Amsterdam, 2000.Google Scholar
Bamber, Nicholas and Kotlarski, Henryk, On interstices of countable arithmetically saturated models of Peano arithmetic. Mathematical Logic Quarterly, vol. 43 (1997), pp. 525–540.CrossRefGoogle Scholar
John Harding, Christopher, Forcing in recursion theory, Ph.D. thesis, Swansea University, Swansea, 1974.Google Scholar
Kaufmann, Matt and Schmerl, James H., Saturation and simple extensions of models of Peano arithmetic. Annals of Pure and Applied Logic, vol. 27 (1984), pp. 109–136.CrossRefGoogle Scholar
Kaufmann, Matt and Schmerl, James H., Remarks on weak notions of saturation in models of Peano arithmetic, this Journal, vol. 52 (1987), pp. 129–148.Google Scholar
Kaye, Richard, Models of Peano arithmetic, Oxford Logic Guides, vol. 15, Oxford Science Publications, Clarendon Press, Oxford, 1991.CrossRefGoogle Scholar
Kaye, Richard, A Galois correspondence for countable recursively saturated models of Peano arithmetic, Automorphisms of first-order structures, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994, pp. 293–312.CrossRefGoogle Scholar
Kaye, Richard, Kossak, Roman and Kotlarski, Henryk, Automorphisms of recursively saturated models of arithmetic. Annals of Pure and Applied Logic, vol. 55 (1991), pp. 67–99.CrossRefGoogle Scholar
Kaye, Richard and Macpherson, Dugald, eds., Automorphisms of first-order structures, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994.CrossRefGoogle Scholar
Knight, Julia F., Minimality and completions of PA, this Journal, vol. 55 (1991), pp. 67–99.Google Scholar
Kossak, Roman, Models with the ω-property, this Journal, vol. 54 (1989), pp. 177–189.Google Scholar
Kossak, Roman, On extensions of models of strong fragments of arithmetic, Proceedings of the American Mathematical Society, vol. 108 (1990), pp. 223–232.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. Special Issue: Models of Arithmetic.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 (2), vol. 52 (1995), pp. 235–244.CrossRefGoogle Scholar
Kossak, Roman and Schmerl, James H., The Structure of Models of Peano Arithmetic, Oxford Logic Guides, vol. 50, Oxford Science Publications, Clarendon Press, Oxford, 2006.CrossRefGoogle Scholar
Kotlarski, Henryk, Automorphisms of countable recursively saturated models of PA: a survey, Special Issue: Models of arithmetic. Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 505–518.CrossRefGoogle Scholar
Lascar, Daniel, The small index property and recursively saturated models of Peano arithmetic, Automorphisms of first-order structures, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1994, pp. 281–292.CrossRefGoogle Scholar
Macintyre, Angus and Marker, David, Degrees of recursively saturated models. Transactions of the American Mathematical Society, vol. 282 (1984), pp. 539–554.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 Schmerl, James H., Automorphism groups of saturated models of Peano Arithmetic, this Journal, vol. 79 (2014), pp. 561–584.Google Scholar
Scott, Dana, Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory (Dekker, J. C. E., editor), American Mathematical Society, Proceedings of Symposia in Pure Mathematics, vol. V, 1962, pp. 117–122.CrossRefGoogle Scholar
Simpson, Stephen G., Subsystems of second order arithmetic, (2nd ed.), Perspectives in Logic, Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Wilmers, George, Some problems in set theory: non-standard models and their applications to model theory, Ph.D. thesis, Oxford University, Oxford, 1975.Google Scholar