Abstract
Abstract
We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than
$\omega $
and that non-standard models of true arithmetic must have Scott rank greater than
$\omega $
. Other than that there are no restrictions. By giving a reduction via
$\Delta ^{\mathrm {in}}_{1}$
bi-interpretability from the class of linear orderings to the canonical structural
$\omega $
-jump of models of an arbitrary completion T of
$\mathrm {PA}$
we show that every countable ordinal
$\alpha>\omega $
is realized as the Scott rank of a model of T.
Publisher
Cambridge University Press (CUP)