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SELF-EMBEDDINGS OF MODELS OF ARITHMETIC; FIXED POINTS, SMALL SUBMODELS, AND EXTENDABILITY
Published online by Cambridge University Press: 22 December 2022
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.
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- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
References
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