Resplendent models

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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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