First some background: Two structures $\mathcal{M}$ and $\mathcal{N}$ are potentially isomorphic if there is a non-empty family of finite partial isomorphisms between them such that

  • for any member of the family $f$ and $x\in\mathcal{M}$, there is $y\in\mathcal{N}$ such that $f\cup\{(x,y)\}$ is in the family.
  • for any member of the family $f$ and $y\in\mathcal{N}$, there is $x\in\mathcal{M}$ such that $f\cup\{(x,y)\}$ is in the family.

Equivalently duplicator has a winning strategy in the $\omega$-length Ehrenfeucht–Fraïssé game on $\mathcal{M}$ and $\mathcal{N}$. Also equivalently $\mathcal{M}$ and $\mathcal{N}$ satisfy the same $\mathcal{L}_{\infty\omega}$ sentences. Also equivalently $\mathcal{M}$ and $\mathcal{N}$ are isomorphic in some forcing extension of the universe.

This implies isomorphism for countable structures but it is a strictly weaker condition for uncountable structures. An easy example is two structureless sets of cardinality $\aleph_0$ and $\aleph_1$. In general $\omega$-categoricity of a theory can be characterized by the condition that any two models (of any cardinality) are potentially isomorphic.

Call a theory '$\kappa$-potentially categorical' if any two models of cardinality $\kappa$ are potentially isomorphic. Since you can always extend a $\kappa$-sized model to an $\omega$-saturated model without increasing the size, this condition is equivalent to the condition that all models of cardinality $\kappa$ are $\omega$-saturated. (Edit: For any $\kappa$, if all models of size $\kappa$ are $\omega$-saturated, then the theory is $\kappa$-potentially categorical. Furthermore I think that: For countable theories and $\kappa\geq 2^{\aleph_0}$ the converse is true; and for countable small theories, i.e. $|S_n(T)|\leq \aleph_0$ for every $n$, the converse is true for arbitrary $\kappa$.) There are two 'trivial' ways to accomplish this. If the theory is already $\kappa$-categorical, then any two models are isomorphic so they are potentially isomorphic. If the theory is $\omega$-categorical then any two models whatsoever are potentially isomorphic, so any two models of cardinality $\kappa$ are potentially isomorphic.

Since countable categoricity and uncountable categoricity are 'orthogonal' properties, potential categoricity can't be equivalent to either of them, but it might be the case that these are the only possibilities.

I haven't been able to construct an example that isn't already categorical in some cardinality. So my question is: does potential categoricity always imply categoricity in some cardinality?

Finally, assuming it doesn't, does potential categoricity in some uncountable cardinality imply potential categoricity in all uncountable cardinalities (i.e. does the analog of Morley's theorem hold)?

Edit 2: The $\kappa$-potentially categorical implies all $\kappa$-sized models are $\omega$-saturated argument goes through the same as the standard argument that countably categorical implies the countable model is saturated: fix an $n$-type $p\in S_n(T)$ and fix an $m$-type $q\in S_m(\overline{a})$ where $\text{tp}(\overline{a})=p$. We can always construct a model, $\mathcal{M}$, of size $\kappa$ which: -realizes the type $p$ and -for every $\overline{b}$ such that $\text{tp}(\overline{b}) = p$, there exists an $\overline{c}$ such that $\text{tp}(\overline{c}/\overline{b})=q$. So in any $\kappa$-sized model $\mathcal{N}$, since $\mathcal{M}$ and $\mathcal{N}$ are potentially isomorphic, by a back-and-forth argument $\mathcal{N}$ must realize $p$ and must also realize $q$ over any realization of $p$. Since we can do this with each type individually $\mathcal{N}$ must be $\omega$-saturated.

This means that if $\aleph_0<\kappa<2^{\aleph_0}$ any $\kappa$-potentially categorical theory must be small (i.e. $|S_n(T)|\leq \aleph_0$ for every $n$) because no $\kappa$-sized model can realize $2^{\aleph_0}$ many types.

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    $\begingroup$ You say "you can always extend a κ-sized model to an ω-saturated model without increasing the size". For every $\kappa$? $\endgroup$ – Primo Petri Jun 27 '17 at 21:33
  • $\begingroup$ Oh right whoops. I think I might have convinced myself that a potentially categorical theory needs to be small (as in all of its type spaces over the empty set are countable), in which case that would be true, right? $\endgroup$ – James Hanson Jun 27 '17 at 23:14
  • $\begingroup$ @Exomnium That's right. Also, if a theory $T$ is potentially categorical in $\kappa$, then $\kappa$ must be at least as large as the number of types over the empty set - so any model of size $\kappa$ can be extended to an $\omega$-saturated one. $\endgroup$ – Alex Kruckman Jun 28 '17 at 8:58
  • $\begingroup$ Related: mathoverflow.net/questions/220850/…. $\endgroup$ – Noah Schweber Jun 28 '17 at 9:07
  • $\begingroup$ (In other words, I think the converse of your observation really is true, i.e. $\kappa$-potentially categorical implies all models of size $\kappa$ are $\omega$-saturated) $\endgroup$ – Alex Kruckman Jun 28 '17 at 9:18

This does not settle your qeustion. However a positive answer to it (i.e. a potentially categorocal theory in some cardinality is already categorical in some other cardinality) implies Kueker's conjecture (if all uncountable models of a theory are $\aleph_0$-saturated, then it is categorical in some cardinality). While Kueker's conjecture is known to be true for many cases (notably stable theories), the general case is I believe still open.

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  • $\begingroup$ That would seem to indicate that the answer to my question is probably not known in full generality. Thank you. $\endgroup$ – James Hanson Jun 30 '17 at 1:08
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    $\begingroup$ In fact, your question is equivalent to a sort of "asymptotic" Kueker's conjecture: If all large enough models of $T$ are $\aleph_0$-saturated, then $T$ is categorical in some cardinality. (Note that if all models of size $\kappa$ are $\aleph_0$-saturated, then so are all models of size $\geq \kappa$.) $\endgroup$ – Alex Kruckman Jun 30 '17 at 14:55

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