Embeddings among $\omega_1$-like models of set theory
This is a talk at the CUNY Set Theory Seminar, October 4, 2013.
Recently Hamkins proved some remarkable theorems about the existence of embeddings between countable models of set theory [1]. We are referring here to embeddings in the model theoretic sense, so that an embedding between two models of set theory is only required to preserve the $\in$-relation and need not possess any further elementarity.
Theorem 1: (Hamkins) Every countable model $\langle M,\in^M\rangle$ of set theory is isomorphic to a submodel of its own constructible universe $\langle L^M,\in^M\rangle$.
Note that if $M\neq L^M$ is transitive, then there cannot even be a $\Delta_0$-elementary embedding $j:M\to L^M$ because transitivity implies that the embedding is cofinal on the ordinals and $\Delta_0$-elementarity implies that $j(L_\alpha)=L_{j(\alpha)}$. On the other hand, it is easy to construct (even definable) embeddings $j:L\to L$ by for instance defining that $j(x)=\{j(y)\mid y\in x\}\cap \{\{x,0\}\}$.
Theorem 2: (Hamkins) For any two countable models of set theory $\langle M,\in^M\rangle$ and $\langle N,\in^N\rangle$, either $M$ is isomorphic to a submodel of $N$ or conversely.
The most remarkable of the embedding results is probably the following theorem extending an already remarkable earlier result of Ressayre. It is a standard observation that every model $M$ of Peano Arithmetic (${\rm PA}$) comes along with a model of finite theory ${\rm HF}^M$ by defining that $m\in ^M n$ whenever the $n^{\text{th}}$-binary digit of $m$ is $1$. Ressayre showed that if $M$ is a nonstandard model of ${\rm PA}$, then for any consistent computably axiomatized theory $T$ extending ${\rm ZF}$ in the language of set theory, the finite set theory model ${\rm HF}^M$ has a submodel $N\models T$ [2]. So that for instance, there is a model of set theory with a measurable cardinal that embeds into a model of finite set theory!
Theorem: (Hamkins) If $M$ is a nonstandard model of ${\rm PA}$, then every countable model of set theory is isomorphic to a submodel of the hereditary finite sets $\langle {\rm HF}^M,\in^M\rangle$.
We should expect these results to mostly fail for uncountable models but the counterexamples are not easy to come by. The best possible place to find a violation of the embedding theorems would be in the class of $\omega_1$-like models. These are almost countable in the sense that an element of an $\omega_1$-like model can have only countably many members, but the model itself has size $\omega_1$, and therefore becomes subject to set theoretic phenomena. More precisely, a model $M$ of set theory is $\omega_1$-like if $|M|=\omega_1$, but for every $b\in M$, the collection of its $\in^M$-members $\{a\in M\mid a\in^M b\}$ is countable. The $\omega_1$-like models of ${\rm ZFC}$ have been extensively studied for a number of reasons. Because they inherit the external structure of $\omega_1$ of the universe in which they reside, set theoretic techniques have been effectively used to produce $\omega_1$-like counterexample models to many fundamental theorems about countable models. These strange $\omega_1$-like models are usually constructed as unions of elementary chains of richly structured countable models and so the constructions themselves often lead to the discovery and exploration of countable models with interesting new properties.
Do there always exists $\omega_1$-like models of ${\rm ZFC}$?
Theorem: (Folklore) The existence of a transitive $\omega_1$-like model of ${\rm ZFC}$ is equiconsistent with an inaccessible cardinal.
Theorem: (Keisler, Morley [3]) If ${\rm ZFC}$ is consistent, then there are $\omega_1$-like models of ${\rm ZFC}$.
In a joint work with Joel David Hamkins and Gunter Fuchs, we show that it is relative consistent to have $\omega_1$-like counterexample models to all the three embedding theorems.
Theorem: (Fuchs, G., Hamkins) Assuming $\diamondsuit$, there are two $\omega_1$-like models $M$ and $N$ of ${\rm ZFC}$ such that neither embeds into the other and there is an $\omega_1$-like model $M$ of ${\rm ZFC}$ and an $\omega_1$-like model $N$ of ${\rm PA}$ such that $M$ does not embed into ${\rm HF}^N$.
Theorem: (Fuchs, G., Hamkins) It is consistent with a Mahlo cardinal that there is a transitive $\omega_1$-like model of ${\rm ZFC}$ that does not embed into its constructible universe $L^M$.
We still do not know whether every ${\rm ZFC}$-universe, which has models of ${\rm ZFC}$, must necessarily have $\omega_1$-like counterexample models to the embedding theorems for countable models.
References
- J. D. Hamkins, “Every countable model of set theory embeds into its own constructible universe,” J. Math. Log., vol. 13, no. 2, pp. 1350006, 27, 2013.
- J. P. Ressayre, “Introduction aux modèles récursivement saturés,” in Séminaire Général de Logique 1983–1984 (Paris, 1983–1984), vol. 27, Paris: Univ. Paris VII, 1986, pp. 53–72.
- H. J. Keisler and M. Morley, “Elementary extensions of models of set theory,” Israel J. Math., vol. 6, pp. 49–65, 1968.