Home

# Automorphism groups of models of Peano arithmetic

## Extract

Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is.

For any structure , let Aut() be its automorphism group. There are groups which are not isomorphic to any model = (N, +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut(N), being a subgroup of Aut((, <)), must be torsion-free. However, as will be proved in this paper, if (A, <) is a linearly ordered set and G is a subgroup of Aut((A, <)), then there are models of PA such that Aut() ≅ G.

If is a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets of A be the basic open subgroups. If ′ is an expansion of , then Aut(′) is a closed subgroup of Aut(). Conversely, for any closed subgroup G ≤ Aut() there is an expansion ′ of such that Aut(′) = G. Thus, if is a model of PA, then Aut() is not only a subgroup of Aut((N, <)), but it is even a closed subgroup of Aut((N, ′)).

There is a characterization, due to Cohn [2] and to Conrad [3], of those groups G which are isomorphic to closed subgroups of automorphism groups of linearly ordered sets.

## References

Hide All
[1]Abramson, F. G. and Harrington, L. A., Models without indiscernibles, this Journal, vol. 43 (1978), pp. 572600.
[2]Cohn, P. M., Groups of order automorphisms of ordered sets, Mathematika, vol. 4 (1957), pp. 4150.
[3]Conrad, P., Right-ordered groups, Michigan Mathematical Journal, vol. 6 (1959), pp. 267275.
[4]Dugas, M. and Göbel, R., All infinite groups are Galois groups over any field, Transactions of the American Mathematical Society, vol. 304 (1987), pp. 355384.
[5]Ehrenfeucht, A., Discernible elements in models of Peano arithmetic, this Journal, vol. 38 (1973), pp. 291292.
[6]Gaifman, H., On models and types of Peano's arithmetic, Annals of Mathematical Logic, vol. 9 (1976), pp. 223306.
[7]Glass, A., Ordered permutation groups, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1981.
[8]Kaye, R., Models of Peano arithmetic, Oxford Logic Guides, Oxford University Press, Oxford, 1991.
[9]Kaye, R., Kossak, R., and Kotlarski, H., Automorphisms of recursively saturated models of arithmetic, Annals of Pure and Applied Logic, vol. 55 (1991), pp. 6799.
[10]Kirby, L. A. S. and Paris, J. B., Initial segments of models of Peano's axioms, Set theory and hierarchy theory V, Bierutowice, Poland, 1976 (Lachlan, A.et al., editor), Lecture Notes in Mathematics, no. 619, 1977, pp. 221226.
[11]Kopytov, V. M. and Medvedev, N. Y., Right-orderable groups, Consultants Bureau, New York, 1996.
[12]Kossak, R., Satisfaction classes and automorphisms of models of PA, Logic colloquium '96, Proceedings of the colloquium held in San Sebastián, Spain, July 9–15, 1996, Springer-Verlag, Berlin, 1998, pp. 159170.
[13]Kossak, R. and Bamber, N., On two questions concerning the automorphism groups of countable recursively saturated models of PA, Archive for Mathematical Logic, vol. 36 (1996), pp. 7379.
[14]Kossak, R. and Schmerl, J. H., Minimal satisfaction classes with an application to rigid models of Peano arithmetic, Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 392398.
[15]Kossak, R. and Schmerl, J. H., Arithmetically saturated models of arithmetic, Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 531546.
[16]Kossak, R. and Schmerl, J. H., The automorphism group of an arithmetically saturated model of Peano arithmetic, Journal of the London Mathematical Society, vol. 52 (1995), no. 2, pp. 235244.
[17]Nešetřil, J. and Rödl, V., Partitions of finite relational and set systems, Journal of Combinatorial Theory A, vol. 22 (1977), pp. 289312.
[18]Robinson, J., Decidability and decision problems in arithmetic, this Journal, vol. 14 (1949), pp. 98114.
[19]Schmerl, J. H., Recursively saturated, rather classless models of Peano arithmetic, Logic year 1979–80, University of Connecticut (Lerman, M.et al., editor), Lecture Notes in Mathematics, no. 859, Springer-Verlag, Berlin, 1981, pp. 268–82.
[20]Smoryński, C. A. and Stavi, J., Cofinal extension preserves recursive saturation, Model theory of algebra and arithmetic, Proceedings, Karpacz, Poland, 1978 (Pacholski, L.et al., editor), Lecture Notes in Mathematics, no. 834, Springer-Verlag, Berlin, 1980, pp. 338345.

# Automorphism groups of models of Peano arithmetic

## Metrics

### Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 0 *

### Abstract viewsAbstract views reflect the number of visits to the article landing page.

Total abstract views: 0 *

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed