# 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 [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

1. John S. Schlipf. A guide to the identification of admissible sets above structures. Ann. Math. Logic 12(2):151--192. MR   bibtex
2. Error: entry with key = schmerl1989:large does not exist
3. James H. Schmerl. Recursively saturated models generated by indiscernibles. Notre Dame J. Formal Logic 26(2):99--105, 1985. www   DOI   MR   bibtex
4. 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
5. 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
6. Grégory Duby. Automorphisms with only infinite orbits on non-algebraic elements. Arch. Math. Logic 42(5):435--447, 2003. www   DOI   MR   bibtex
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