# Resplendent models

## 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 : 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 $(M,R_0,R_1,\dots)$ .

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 , 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 $PA$ . 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 $M^+$?