Difference between revisions of "Uncountable models with interesting second-order properties"
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== Friedman's 14th problem == | == Friedman's 14th problem == | ||
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Gaifman <cite> gaifman1976:models </cite> and Knight <cite> knight1976:hanf </cite> independently showed that there are Jonsson models of $PA$. | Gaifman <cite> gaifman1976:models </cite> and Knight <cite> knight1976:hanf </cite> independently showed that there are Jonsson models of $PA$. | ||
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| − | Jonsson models $M$ of $PA$ of cardinality $\aleph_1$ are either $\aleph_1-like | + | 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 $|\{{\rm tp}(a): a\in M\}|=\aleph_0$? | 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 <cite> kossak1995:four </cite>. | 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 <cite> kossak1995:four </cite>. | ||
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== Rigid models == | == Rigid models == | ||
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Problem: Are there rigid recursively saturated $M\models PA$ such that ${\rm cf}(M)=\aleph_0$? | Problem: Are there rigid recursively saturated $M\models PA$ such that ${\rm cf}(M)=\aleph_0$? | ||
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| + | == Lofty models == | ||
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| + | 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 <cite> kaufmannschmerl1984:saturation </cite> (and also in <cite> kaufmannschmerl1987:remarks </cite>), | ||
| + | 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 | ||
| + | <cite> kaufmannschmerl1987:remarks </cite>, but the question remains open. | ||
{{References}} | {{References}} | ||
Latest revision as of 12:39, 18 February 2013
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
