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INTERPRETATIONS BETWEEN ω-LOGIC AND SECOND-ORDER ARITHMETIC

  • RICHARD KAYE (a1)
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.

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References
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[1]Barwise, Jon and Schlipf, John, An introduction to recursively saturated and resplendent models, this Journal, vol. 41 (1976), no. 2, pp. 531536.
[2]Engström, Fredrik and Kaye, Richard, Transplendent models: Expansions omitting a type. Notre Dame Journal of Formal Logic, vol. 53 (2012), no. 3, pp. 413428.
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[7]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.
[8]Schmerl, James H., A reflection principle and its applications to nonstandard models, this Journal, vol. 60 (1995), no. 4, pp. 11371152.
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[10]Smith, Stuart T., Extendible sets in Peano arithmetic. Transactions of the American Mathematical Society, vol. 316 (1989), no. 1, pp. 337367.
[11]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.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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