Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T22:39:58.879Z Has data issue: false hasContentIssue false
  • English
  • Français

Recursively saturated nonstandard models of arithmetic

Published online by Cambridge University Press:  12 March 2014

C. Smoryński
C. Smoryński*
Affiliation:
Universität Heidelberg, 6900 Heidelberg 1, Federal Republic of Germany
*
429 S. Warwick, Westmont, Illinois 60559
Get access Rights & Permissions [Opens in a new window]

Extract

Through the ability of arithmetic to partially define truth and the ability of infinite integers to simulate limit processes, nonstandard models of arithmetic automatically have a certain amount of saturation: Any encodable partial type whose formulae all fall into the domain of applicability of a truth definition must, by finite satisfiability and Overspill, be nonstandard-finitely satisfiable—whence realized. This fact was first exploited by A. Robinson, who in Robinson [1963] cited the unrealizability in a given model of a certain encodable partial type to prove Tarski's Theorem on the Undefinability of Truth. A decade later, H. Friedman brought this phenomenon to the public's attention by using it to establish impressive embeddability criteria for countable nonstandard models of arithmetic. Subsequently, Wilkie considered models expandable to “strong theories” and, among such models, complemented Friedman's embeddability criteria with elementary embeddability and isomorphism criteria. Oddly enough, the fact that some kind of saturation property was being employed was not explicitly acknowledged in any of this work.

It is in the unpublished dissertation of Wilmers that these submerged saturation properties first surfaced. [As I have only seen accounts of it (most notably Murawski [1976/1977]) and not the dissertation itself, what I have to say about it will not quite be accurate. (Indeed, the referee has refuted, without providing alternate information, every conjecture I have made about the contents of this thesis.)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 2 , June 1981 , pp. 259 - 286
Copyright
Copyright © Association for Symbolic Logic 1981

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

Barwise, J. and Schlipf, J. [1975] On recursively saturated models of arithmetic, Model theory and algebra: A memorial tribute to Abraham Robinson(Saracino, D.H. and Weispfenning, V.B., Editors), Springer-Verlag, Heidelberg.Google Scholar
Blass, A. [1972] The intersection of nonstandard models of arithmetic, this Journal, vol. 37, pp. 103–106.Google Scholar
Ehrenfeucht, A. and Kreisel, G. [1966] Strong models for arithmetic, Bulletin de l'Académie Polonaise des Sciences, vol. 14, pp. 107–110.Google Scholar
Feferman, S. [1960] Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae, vol. 49, pp. 35–92.CrossRefGoogle Scholar
Feferman, S. [1977] Theories of finite type related to mathematical practice, Handbook of mathematical logic (Barwise, J., Editor), North-Holland, Amsterdam.Google Scholar
Friedman, H. [1973] Countable models of set theories, Cambridge Summer School in Mathematical Logic (Mathias, A.R.D. and Rogers, H., Editors), Springer-Verlag, Heidelberg.Google Scholar
Friedman, H. [1977] Set theoretic foundations for constructive analysis, Annals of Mathematics, vol. 105, pp. 1–28.CrossRefGoogle Scholar
Gaifman, H. [1972] A note on models and submodels of arithmetic, Conference in Mathematical Logic, London, 1970 (Hodges, W., Editor), Springer-Verlag, Heidelberg.Google Scholar
Gaifman, H. [1976] Models and types of Peano's arithmetic, Annals of Mathematical Logic, vol. 9, pp. 223–306.CrossRefGoogle Scholar
Guaspari, D. [1979] Partially conservative extensions of arithmetic, Transactions of the American Mathematical Society, vol. 254, pp. 47–68.CrossRefGoogle Scholar
Hájek, P. [1972] On interpretability in set theories II, Commentationes Mathematicae Universitatis Carolinae, vol. 13, pp. 445–455.Google Scholar
Hasenjäger, G. [1953] Eine Bemerkung zu Henkin's Beweis für die Vollständigkeit des Prädikatenkalküls der ersten Stufe, this Journal, vol. 18, pp. 42–48.Google Scholar
Hilbert, D. and Bernays, P. [1939] Grundlagen der Mathematik. II, Springer-Verlag, Berlin.Google Scholar
Jensen, D. and Ehrenfeucht, A. [1976] Some problem in elementary arithmetics, Fundamenta Mathematicae, vol. 92, pp. 223–245.CrossRefGoogle Scholar
Kirby, L. and Paris, J. [1977] Initial segments of models of Peano's axioms, Set theory and hierarchy theory. V (Lachlan, A., Srebrny, M. and Zarach, A., Editors), Springer-Verlag, Heidelberg.Google Scholar
Kleene, S. [1952] Introduction to metamathematics, Van Nostrand, New York.Google Scholar
Lesan, H. [1978] Models of arithmetic, Dissertation, University of Manchester.Google Scholar
Lipshitz, L. [1979] Diophantine correct models of arithmetic, Proceedings of the Amerian Mathematical Society, vol. 73, pp. 107–108.CrossRefGoogle Scholar
MacDowell, R. and Specker, E. [1961] Modelle der Arithmetik, Infinitistic methods, Pergamon Press, London.Google Scholar
Manevitz, L. [1976] Internal end-extensions of Peano arithmetic and a problem of Gaifman, Journal of the London Mathematical Society, vol. 13, pp. 80–82.CrossRefGoogle Scholar
McAloon, K. [1978] Completeness theorems, incompleteness theorems and models of arithmetic, Transactions of the American Mathematical Society, vol. 239, pp. 253–277.CrossRefGoogle Scholar
Mlček, J. [1978] A note on cofinal extensions and segments, Commentationes Mathematicae Universitatis Carolinae, vol. 19, pp. 727–742.Google Scholar
Mostowski, A. [1952] On models of axiomatic systems, Fundamenta Mathematicae, vol. 39, pp. 133–158.CrossRefGoogle Scholar
Murawski, R. [1976/1977] On expandability of models of Peano arithmetic. I, II, III, Studia Logica, vol. 35, pp. 409–419 and 421–431; vol. 36, pp. 181–188; correction: Studia Logica, vol. 36.CrossRefGoogle Scholar
Nadel, M. [A] On a problem of MacDowell and Specker (to appear).Google Scholar
Orey, S. [1961] Relative interpretations, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 7, pp. 146–153.CrossRefGoogle Scholar
Paris, J. [1978] Some independence results for Peano arithmetic, this Journal, vol. 43, pp. 725–731.Google Scholar
Parsons, C. [1970] On a number-theoretic choice schema and its relation to induction, Intuitionism and proof theory (Kino, A., Myhill, J. and Vesley, R., Editors), North-Holland, Amsterdam.Google Scholar
Parsons, C. [1972] On n-quantifier induction, this Journal, vol. 37, pp, 466–482.Google Scholar
Ressayre, J. P. [1977] Models with compactness properties relative to an admissible language, Annals of Mathematical Logic, vol. 11, pp. 31–55.CrossRefGoogle Scholar
Robinson, A. [1963] On languages which are based on nonstandard arithmetic, Nagoya Mathematical Journal, vol. 22, pp. 83–117; reprinted in Selected papers of Abraham Robinson. II; Nonstandard analysis and philosophy (W.A.J. Luxemburg and S. Körner, Editors), Yale University, New Haven.CrossRefGoogle Scholar
Ryll-Nardzewski, C. [1952] The role of the axiom of induction in elementary arithmetic, Fundamenta Mathematicae, vol. 39, pp. 239–263.CrossRefGoogle Scholar
Schlipf, J. [1977] A guide to the identification of admissible sets above structures, Annals of Mathematical Logic, vol. 12, pp. 151–192.CrossRefGoogle Scholar
Scott, D. [1962] Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory (Dekker, J.C.E., Editor), American Mathematical Society, Providence, R.I.Google Scholar
Shelah, S. [1978] Models with second order properties. II, Trees with no undefined branches, Annals of Mathematical Logic, vol. 14, pp. 73–87.CrossRefGoogle Scholar
Smoryński, C. [1973] Applications of Kripke models, Metamathematical investigation of intuitionistic arithmetic and analysis (Troelstra, A.S., Editor), Springer-Verlag, Heidelberg.Google Scholar
Smoryński, C. and Stavi, J. [A] Cofinal extension preserves recursive saturation, The model theory of algebra and arithmetic (Pacholski, L. and Wilkie, A., Editors), Springer-Verlag, Heidelberg (to appear).Google Scholar
Wilkie, A. [A] On the arithmetical parts of strong theories (unpublished).Google Scholar
Wilkie, A. [1975] On models of arithmetic—answers to two problems raised by H. Gaifman, this Journal, vol. 40, pp. 41–47.Google Scholar
Wilkie, A. [1977] On the theories of end-extensions of models of arithmetic, Set theory and hierarchy theory. V (Lachlan, A., Srebrny, M. and Zarach, A., Editors), Springer-Verlag, Heidelberg.Google Scholar