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The following arose from a question in model theory (specifically in the model theory of modules). Fix an arity $k$. Let $[\mathbb{Q}]^k$ denote the set of all subsets of $\mathbb{Q}$ of cardinality $ …

Here is a partial answer: consistently, the generalization can fail for all uncountable $\kappa$. Namely: Suppose $\mathbb{V} = \mathbb{L}$ and let $\kappa$ be any uncountable cardinal. Let $\mathca …

Say that a complete theory $T$ is a limit of stable theories if for every $\phi \in T$ there is a stable completion of $\{\phi\}$. (Equivalently, $T$ is the ultraproduct of stable theories.) Question …

The property you are asking for is a very strong condition on $T$. Let met try to rephrase the question more carefully: The set of countable $\Sigma$-structures with universe $\omega$ is naturally a …

Yes. Let $T$ be the theory of two disjoint groups, in the language $(\cdot_1, \cdot_2, U_1, U_2)$. Note that if $(G_1, G_2) \models T$ then the group operation on $G_1 \times G_2$ is definable without …

The answer to the question is yes, assuming $VP$ holds and thus, in the terminology from your earlier question, that $VP(\mathcal{L}_{\omega_1 \omega})$ holds. Namely let $T$ be any unstable theory wi …

It is a fact (following from the Ehrenfeucht–Mostowski theorem) that for every complete theory $T$ and for every $\lambda \geq |T|$, there is $M \models T$ with $|M| = \lambda$ and $M$ having $2^\lamb …

For an example of a super-stable theory failing the property, you can take infinitely many unary predicates (so $Th(2^\omega, U_n: n \in \omega)$ where $U_n(\eta)$ holds iff $\eta(n) = 1$). Then if $\ …

Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its …