Abstract
Abstract
In this paper we will show that for every cut I of any countable nonstandard model
$\mathcal {M}$
of
$\mathrm {I}\Sigma _{1}$
, each I-small
$\Sigma _{1}$
-elementary submodel of
$\mathcal {M}$
is of the form of the set of fixed points of some proper initial self-embedding of
$\mathcal {M}$
iff I is a strong cut of
$\mathcal {M}$
. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model
$\mathcal {M}$
of
$ \mathrm {I}\Sigma _{1} $
. In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of
$ \mathrm {I}\Sigma _{1} $
to larger models.
Publisher
Cambridge University Press (CUP)