# Resplendent models

## Contents

## Chronically resplendent models

This problem, and the next, are not specifically about models of $PA$, but both questions are interesting in the context of $PA$.

Every countable resplendent model is *chronically resplendent*, which means that the expansions given by resplendency can be also made resplendent.

Problem [1]: Is every resplendent model chronically resplendent?

## Totally resplendent models

A model $M$ is *totally resplendent* if there are countably many relations $R_0, R_1,\dots$ on $M$ such that each expansion $(M,R_0,R_1,\dots, R_{n-1})$ is resplendent and moreover if $(M,R_0,R_1,\dots)\models\exists R\ \varphi (\bar a, R)$, then $(M,R_0,R_1,\dots)\models \varphi (\bar a, R)$ for some $R$ parametrically definable in $(M,R_0,R_1,\dots)$ [2].

Every countable resplendent model is totally resplendent.

Problem: Is every resplendent model totally resplendent?

## A converse to Schmerl's theorem?

By a theorem of Schmerl [3], every countable recursively saturated model of $PA$ is generated by a set of indiscernibles of any coutnable order type without a last element.

Problem: Suppose $M$ is a countable, tall model of $PA$, and suppose $M$ is generated by sets of indiscernibles of two different order types. Is $M$ recursively saturated?

## Gentle expansions

This problem is not directly about models of PA, but is motivated by results concerning maximal automorphisms, first proved in the context of arithmetically saturated models of $PA$ [4]. For results on maximal automorphisms in general setting see [5, 6]

An expansion $M^+$ of a structure $M$ is *gentle* if all algebraic elements of $M^+$ are
already algebraic in $M$.

Problem: Let $M$ be a countable resplendent model. Can $M$ always be gently expanded to a linearly ordered
resplendent $M^+$?

## References

- John S. Schlipf.
*A guide to the identification of admissible sets above structures.*Ann. Math. Logic 12(2):151--192. MR bibtex -
**Error:**entry with key = schmerl1989:large does not exist - James H. Schmerl.
*Recursively saturated models generated by indiscernibles.*Notre Dame J. Formal Logic 26(2):99--105, 1985. www DOI MR bibtex - Richard Kaye, Roman Kossak and Henryk Kotlarski.
*Automorphisms of recursively saturated models of arithmetic.*Ann. Pure Appl. Logic 55(1):67--99, 1991. www DOI MR bibtex - Friederike Körner.
*Automorphisms moving all non-algebraic points and an application to NF.*J. Symbolic Logic 63(3):815--830, 1998. www DOI MR bibtex - Grégory Duby.
*Automorphisms with only infinite orbits on non-algebraic elements.*Arch. Math. Logic 42(5):435--447, 2003. www DOI MR bibtex