Incomparable $\omega_1$-like models of set theory

This is a talk at the Connecticut Logic Seminar, March 31, 2014.
Slides

To paraphrase Roman Kossak, $\omega_1$-like models bridge the gap between countable and uncountable structures. A model $\mathcal M=\langle M,+,\cdot,<,0,1\rangle$ of ${\rm PA}$ is $\omega_1$-like if it has size $\omega_1$, but all its proper $<$-initial segments are countable. A model $\mathcal M=\langle M,\in\rangle$ of ${\rm ZFC}$ is $\omega_1$-like if it has size $\omega_1$, but all its rank initial segments $V_\alpha^{\mathcal M}$ are countable. Because the $\omega_1$-like models are locally countably one could naively imagine that theorems about countable models would generalize to $\omega_1$-like models. Indeed, it is nearly always the opposite that is the case. The entire set theoretic baggage of $\omega_1$ gets implicitly built into an $\omega_1$-like model. Suddenly the techniques of clubs, stationary sets, reflection, the $\diamondsuit$-principle, etc. can be applied to modifying and controlling purely model theoretic properties. If you want to violate a fundamental theorem about countable models in an uncountable case, your best bet is usually an $\omega_1$-like model. Alternatively, if you seek an uncountable model with some property that becomes interesting only in the uncountable context, again $\omega_1$-like models are usually the way to go. Let's mention a few notable $\omega_1$-like models.

First some positive constructions. A model of cardinality $\kappa$ is Jónsson if it has no proper elementary substructures of cardinality $\kappa$. Julia Knight showed that there is a Jónsson $\omega_1$-like model of ${\rm ZFC}$ (or ${\rm PA}$) [1]. Ali Enayat calls a model Leibnizian if it has no indiscernibles, that is, every element has a unique type. This is meant to recall Leibniz who postulated the Identity of Indiscernibles principle stating that no two distinct objects should share all the same properties. It is not difficult to construct countable Leibnizian models because any point-wise definable countable model is Leibnizian. Enayat showed that there is an $\omega_1$-like Leibnizian model of ${\rm ZFC}$ and it is still open whether one can have a Leibnizian model of size continuum if ${\rm CH}$ fails [2].

Now to the counterexamples. An end-extension of a model of ${\rm ZFC}$ adds only new elements of higher rank than all ordinals of the original model. By a classical theorem of Keisler and Morley, every countable model of ${\rm ZFC}$ has an elementary end-extension [3]. But Kaufmann showed that there is an $\omega_1$-like model of ${\rm ZFC}$ that doesn't have an elementary end-extension [bibcite key=kaufmann:omega1likemodels]. An elegant result from models of ${\rm PA}$ is the complete classification of countable computably saturated models using two invariants: theory and standard system. The standard system ${\rm SSy}(\mathcal M)$ of $\mathcal M\models{\rm PA}$ is the collection of subsets of $\mathbb N$ that arise as intersections of its initial segment $\mathbb N$ with definable subsets of $\mathcal M$. Two countable recursively saturated models of ${\rm PA}$ are isomorphic if and only if they share the same theory and standard system. The $\omega_1$-like models have a counterexample. Kossak constructed two recursively saturated $\omega_1$-like models with the same theory and standard system such that neither embeds into the other [4]. This talk is about another counterexample in the class of $\omega_1$-like models to a theorem about countable models.

Hamkins showed recently that given any two countable models of ${\rm ZFC}$ one of them embeds into the other [5]. Indeed, whether a countable model of ${\rm ZFC}$ embeds into another countable model is completely determined by whether its ordinals embed. The intuition from the very beginning was that the result would fail for $\omega_1$-like models. In a joint work with Gunter Fuchs and Joel Hamkins, we show that [6]:

Theorem: Assuming $\diamondsuit$, if ${\rm ZFC}$ is consistent, then there is a collection $\mathcal C$ of the maximum possible size $2^{\omega_1}$ of $\omega_1$-like models of ${\rm ZFC}$ such that there is no embedding between any pair of models in $\mathcal C$.

For more on this project read here.

References

  1. J. F. Knight, “Hanf numbers for omitting types over particular theories,” J. Symbolic Logic, vol. 41, no. 3, pp. 583–588, 1976.
  2. A. Enayat, “Leibnizian models of set theory,” J. Symbolic Logic, vol. 69, no. 3, pp. 775–789, 2004.
  3. H. J. Keisler and M. Morley, “Elementary extensions of models of set theory,” Israel J. Math., vol. 6, pp. 49–65, 1968.
  4. R. Kossak, “Recursively saturated $\omega_1$-like models of arithmetic,” Notre Dame J. Formal Logic, vol. 26, no. 4, pp. 413–422, 1985.
  5. 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.
  6. G. Fuchs, V. Gitman, and J. D. Hamkins, “Incomparable $\omega_1$-like models of set theory,” MLQ Math. Log. Q., vol. 63, no. 1-2, pp. 66–76, 2017.