1 results
SELF-EMBEDDINGS OF MODELS OF ARITHMETIC; FIXED POINTS, SMALL SUBMODELS, AND EXTENDABILITY
- Part of
-
- Journal:
- The Journal of Symbolic Logic / Volume 89 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 22 December 2022, pp. 1044-1066
- Print publication:
- September 2024
-
- Article
-
- You have access
- HTML
- Export citation
-
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.
