## Journal of Symbolic Logic

### On External Scott Algebras in Nonstandard Models of Peano Arithmetic

#### Abstract

We prove that a necessary and sufficient condition for a countable set $\mathscr{L}$ of sets of integers to be equal to the algebra of all sets of integers definable in a nonstandard elementary extension of $\omega$ by a formula of the $\mathrm{PA}$ language which may include the standardness predicate but does not contain nonstandard parameters, is as follows: $\mathscr{L}$ is closed under arithmetical definability and contains $0^{(\omega)}$, the set of all (Godel numbers of) true arithmetical sentences. Some results related to definability of sets of integers in elementary extensions of $\omega$ are included.

#### Article information

Source
J. Symbolic Logic, Volume 61, Issue 2 (1996), 586-607.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745016

Mathematical Reviews number (MathSciNet)
MR1394616

Zentralblatt MATH identifier
0859.03034

JSTOR