# Uncountable models with interesting second-order properties

## Contents

## Friedman's 14th problem

Let $T$ be a completion of PA. Let ${\rm Ot}(T)$ be the spectrum of order types of nonstandard models of $T$. In [1] Harvey Friedman asked: Does ${\rm Ot(T)}$ depend on $T$?

Shelah has recently announced that the answer is negative, but there is no paper available yet. See Sh[924] for an interesting effort in the opposite direction.

## Rather classless models

$X\subseteq M\models PA$ is a *class* if for all $a\in M$, $\{x\in X: x<a\}$ is definable (coded) in $M$.
$M$ is *rather classless* if each class of $M$ is definable.

Using the fact that every model of $PA$ has a conservative elementary end extension [2], Schmerl [3] proved that for each cardinal $\kappa$ such that ${\rm cf}(\kappa)>\aleph_0$, there are $\kappa$-like rather classless models of $PA$. A model is *$\kappa$-like* if it is of cardinality $\kappa$ and each of its proper initial segments is of smaller cardinality.

Kaufmann [4], assuming $\lozenge$, proves that there are recursively saturated $\aleph_1$-like rather classless models. Later Shelah showed that $\lozenge$ can be eliminated from the proof. Nevertheless one can still ask, as Hodges did in [5]: Prove the existence of rather classless recursively saturated models of $PA$ in cardinality $\aleph_1$ without assuming diamond at any stage of the proof.

## Jonsson models

A model $\mathfrak B$ is *Jonsson* if $|{\mathfrak B}|>\aleph_0$ and for every ${\mathfrak A}\prec {\mathfrak B}$, if $|{\mathfrak A}|=|{\mathfrak B}|$, then ${\mathfrak A}={\mathfrak B}$.

Gaifman [2] and Knight [6] independently showed that there are Jonsson models of $PA$.
.
Jonsson models $M$ of $PA$ of cardinality $\aleph_1$ are either $\aleph_1$-like or are *short*. A model $M$ of $PA$ is short there is an $a\in M$ such that the Skolem closure of $a$ is cofinal in $M$. Each known Jonsson model realizes uncountably many complete types.

Kossak has asked: Is there an $\aleph_1$-like Jonsson model $M\models PA$ such that $|\{{\rm tp}(a): a\in M\}|=\aleph_0$?

If $M\models PA$ is $\aleph_1$-like and recursively saturated, then $|\{{\rm tp}(a): a\in M\}|=\aleph_0$, but $M$ is not Jonsson. Therefore, another related question is: Is there a *weakly Jonsson model* $M\models PA$, i.e. a recursively saturated model $M\models PA$ such that for every recursively saturated $K\prec M$, if $|K|=|M|$, then $K=M$? The problem was posed in [7].

## Rigid models

There are $\aleph_1$-like rigid models of $PA$. Two different constructions are given in [8] and [9]

Problem: Are there rigid recursively saturated $M\models PA$ such that ${\rm cf}(M)=\aleph_0$?

## Lofty models

A model $M$ of PA is **lofty** if there is $e \in M$ such that whenever $a \in M$ and
$\Phi = \{\varphi_i(a,x) : i < \omega\}$ is a recursive set of formulas that is finitely satisfiable
in $M$, then there is a definable $D \subseteq M$ such that $M \models |D| \leq e$ and
$\Phi \cup \{x \in D\}$ is finitely satisfiable in $M$. This notion was defined and studied by
Kaufmann and Schmerl in [10] (and also in [11]),
where the following was proved:
If $M$ is a countable model of PA, then $M$ is lofty iff $M$ has a simple elementary extension (i.e. generated by a single element) that is recursively saturated. The question was asked whether
the modifier "countable" could be eliminated. Further progress was made in
[11], but the question remains open.

## References

- Harvey Friedman.
*One hundred and two problems in mathematical logic.*J. Symbolic Logic 40:113--129, 1975. MR bibtex - Haim Gaifman.
*Models and types of Peano's arithmetic.*Ann. Math. Logic 9(3):223--306, 1976. MR bibtex - James H. Schmerl.
*Recursively saturated, rather classless models of Peano arithmetic.*Logic Year 1979--80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80)859:268--282, Berlin, 1981. MR bibtex - Matt Kaufmann.
*A rather classless model.*Proc. Amer. Math. Soc. 62(2):330--333, 1977. MR bibtex - Wilfrid Hodges.
*Building models by games.*Vol. 2, Cambridge University Press, Cambridge, 1985. MR bibtex - Julia F. Knight.
*Hanf numbers for omitting types over particular theories.*J. Symbolic Logic 41(3):583--588, 1976. MR bibtex - Roman Kossak.
*Four problems concerning recursively saturated models of arithmetic.*Notre Dame J. Formal Logic 36(4):519--530, 1995. (Special Issue: Models of arithmetic) www DOI MR bibtex - Roman Kossak and James H. Schmerl.
*Minimal satisfaction classes with an application to rigid models of Peano arithmetic.*Notre Dame J. Formal Logic 32(3):392--398, 1991. www DOI MR bibtex - Roman Kossak and Henryk Kotlarski.
*Game approximations of satisfaction classes and the problem of rather classless models.*Z. Math. Logik Grundlag. Math. 38(1):21--26, 1992. www DOI MR bibtex - Matt Kaufmann and James H. Schmerl.
*Saturation and simple extensions of models of Peano arithmetic.*Ann. Pure Appl. Logic 27(2):109--136, 1984. www DOI MR bibtex - Matt Kaufmann and James H. Schmerl.
*Remarks on weak notions of saturation in models of Peano arithmetic.*J. Symbolic Logic 52(1):129--148, 1987. www DOI MR bibtex