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ENAYAT MODELS OF PEANO ARITHMETIC
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- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 4 / December 2018
- Published online by Cambridge University Press:
- 21 December 2018, pp. 1501-1511
- Print publication:
- December 2018
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- Article
- Export citation
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Simpson [6] showed that every countable model
${\cal M} \models PA$ has an expansion 
$\left( {{\cal M},X} \right) \models P{A^{\rm{*}}}$ that is pointwise definable. A natural question is whether, in general, one can obtain expansions of a nonprime model in which the definable elements coincide with those of the underlying model. Enayat [1] showed that this is impossible by proving that there is 
${\cal M} \models PA$ such that for each undefinable class X of 
${\cal M}$, the expansion 
$\left( {{\cal M},X} \right)$ is pointwise definable. We call models with this property Enayat models. In this article, we study Enayat models and show that a model of 
$PA$ is Enayat if it is countable, has no proper cofinal submodels and is a conservative extension of all of its elementary cuts. We then show that, for any countable linear order γ, if there is a model 
${\cal M}$ such that 
$Lt\left( {\cal M} \right) \cong \gamma$, then there is an Enayat model 
${\cal M}$ such that 
$Lt\left( {\cal M} \right) \cong \gamma$.
