Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-04T18:59:40.225Z Has data issue: false hasContentIssue false

INTERPRETATIONS BETWEEN ω-LOGIC AND SECOND-ORDER ARITHMETIC

Published online by Cambridge University Press:  18 August 2014

RICHARD KAYE*
Affiliation:
SCHOOL OF MATHEMATICS, UNIVERSITY OF BIRMINGHAM, B15 2TT, UKE-mail: [email protected]

Abstract

This paper addresses the structures (M, ω) and (ω, SSy(M)), where M is a nonstandard model of PA and ω is the standard cut. It is known that (ω, SSy(M)) is interpretable in (M, ω). Our main technical result is that there is an reverse interpretation of (M, ω) in (ω, SSy(M)) which is ‘local’ in the sense of Visser [11]. We also relate the model theory of (M, ω) to the study of transplendent models of PA [2].

This yields a number of model theoretic results concerning the ω-models (M, ω) and their standard systems SSy(M, ω), including the following.

$\left( {M,\omega } \right) \prec \left( {K,\omega } \right)$ if and only if $M \prec K$ and $\left( {\omega ,{\rm{SSy}}\left( M \right)} \right) \prec \left( {\omega ,{\rm{SSy}}\left( K \right)} \right)$.

$\left( {\omega ,{\rm{SSy}}\left( M \right)} \right) \prec \left( {\omega ,{\cal P}\left( \omega \right)} \right)$ if and only if $\left( {M,\omega } \right) \prec \left( {{M^{\rm{*}}},\omega } \right)$ for some ω-saturated M*.

$M{ \prec _{\rm{e}}}K$ implies SSy(M, ω) = SSy(K, ω), but cofinal extensions do not necessarily preserve standard system in this sense.

• SSy(M, ω)=SSy(M) if and only if (ω, SSy(M)) satisfies the full comprehension scheme.

• If SSy(M, ω) is uniformly defined by a single formula (analogous to a β function), then (ω, SSy(M, ω)) satisfies the full comprehension scheme; and there are models M for which SSy(M, ω) is not uniformly defined in this sense.

Type
Articles
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

Barwise, Jon and Schlipf, John, An introduction to recursively saturated and resplendent models, this Journal, vol. 41 (1976), no. 2, pp. 531536.Google Scholar
Engström, Fredrik and Kaye, Richard, Transplendent models: Expansions omitting a type. Notre Dame Journal of Formal Logic, vol. 53 (2012), no. 3, pp. 413428.Google Scholar
Kanovei, Vladimir, On external Scott algebras in nonstandard models of Peano arithmetic, this Journal, vol. 61 (1996), no. 2, pp. 586607.Google Scholar
Kaye, Richard, Models of Peano arithmetic, Oxford University Press, Oxford, 1991.Google Scholar
Kossak, Roman and Schmerl, James H., The structure of models of Peano arithmetic, Oxford Logic Guides, vol. 50, The Clarendon Press, Oxford University Press, Oxford, 2006, Oxford Science Publications.CrossRefGoogle Scholar
Ressayre, J. P., Models with compactness properties relative to an admissible language. Annals of Mathematical Logic, vol. 11 (1977), no. 1, pp. 3155.Google Scholar
Kaye, Richard, Kossak, Roman, and Wong, Tin Lok, Adding standardness to nonstandard arithmetic. New studies in weak arithmetics (Cégielski, P., Cornaros, Ch., and Dimitracopoulos, C., editors), CSLI Lecture notes number 211, CSLI Publications, Stanford, 2014, pp. 179197.Google Scholar
Schmerl, James H., A reflection principle and its applications to nonstandard models, this Journal, vol. 60 (1995), no. 4, pp. 11371152.Google Scholar
Simpson, Stephen G., Subsystems of second order arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009.Google Scholar
Smith, Stuart T., Extendible sets in Peano arithmetic. Transactions of the American Mathematical Society, vol. 316 (1989), no. 1, pp. 337367.Google Scholar
Visser, Albert, Categories of theories and interpretations, Logic in Tehran, Lecture notes in logic, vol. 26, The Association for Symbolic Logic, La Jolla, CA, 2006, pp. 284341.Google Scholar