# Uncountable models with interesting second-order properties

## 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

1. Harvey Friedman. One hundred and two problems in mathematical logic. J. Symbolic Logic 40:113--129, 1975. MR   bibtex
2. Haim Gaifman. Models and types of Peano's arithmetic. Ann. Math. Logic 9(3):223--306, 1976. MR   bibtex
3. 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
4. Matt Kaufmann. A rather classless model. Proc. Amer. Math. Soc. 62(2):330--333, 1977. MR   bibtex
5. Wilfrid Hodges. Building models by games. Vol. 2, Cambridge University Press, Cambridge, 1985. MR   bibtex
6. Julia F. Knight. Hanf numbers for omitting types over particular theories. J. Symbolic Logic 41(3):583--588, 1976. MR   bibtex
7. 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
8. 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
9. 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
10. 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
11. 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
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