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On external Scott algebras in nonstandard models of Peano arithmetic

  • Vladimir Kanovei (a1)


We prove that a necessary and sufficient condition for a countable set of sets of integers to be equal to the algebra of all sets of integers definable in a nonstandard elementary extension of ω by a formula of the PA language which may include the standardness predicate but does not contain nonstandard parameters, is as follows: is closed under arithmetical definability and contains 0(ω) the set of all (Gödel numbers of) true arithmetical sentences.

Some results related to definability of sets of integers in elementary extensions of ω are included.



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[1]Harrington, L., The constructible reals can be anything, a preprint dated May 1974 with several addendums dated up to 10 1975.
[2]Hinman, P., Recursion theoretic hierarchies, Springer-Verlag, Berlin, 1978.
[3]Kanovei, V., The set of all analytically definable sets of natural numbers can be defined analytically, Mathematics of the USSR-Izvestiya, vol. 15 (1980), pp. 469500.
[4]Kaye, R. W., Models of Peano arithmetic, Oxford logic guides, vol. 15, Oxford Science Publications, 1991.
[5]Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory, Proceedings of the Fifth Symposium in Pure Mathematics of the American Mathematical Society, 1962, pp. 117121.


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