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A note on standard systems and ultrafilters

  • Fredrik Engström (a1)

Let (M, )⊨ ACA0 be such that , the collection of all unbounded sets in , admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in such that M thinks T is consistent. We prove that there is an end-extension NT of M such that the subsets of M coded in N are precisely those in . As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory T extending first order arithmetic is the standard system of a recursively saturated model of T.

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[1979] David Guaspari , Partially conservative extensions of arithmetic, Transactions of the American Mathematical Society, vol. 254 (1979), pp. 4768.

[1984] Laurence Kirby , Ultrafilters and types on models of arithmetic, Annals of Pure and Applied Logic, vol. 27 (1984), no. 3, pp. 215252.

[1962] Dana Scott , Algebras of sets binumerable in complete extensions of arithmetic, Proceedings of symposia in pure mathematics, vol. V, American Mathematical Society, Providence, R.I., 1962, pp. 117121.

[1999] Stephen G. Simpson , Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999.

[1984] Craig Smoryński , Lectures on nonstandard models of arithmetic, Logic colloquium '82 (Florence, 1982), Studies in Logic and the Foundations of Mathematics, vol. 112, North-Holland, Amsterdam, 1984, pp. 170.

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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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