# Categoricity of theory expressing a certain kind of equivalence relation

Consider the language $$\mathcal{L}$$ of first-order logic equipped with a single binary relation symbol $$R$$. Let $$\Gamma$$ denote the theory expressing that $$R$$ is an equivalence relation such that there is exactly one equivalence class of size $$n$$ for each $$n\in \mathbb{Z}_{>0}$$.

Is it the case that $$\Gamma$$ is $$\kappa$$-categorical for every uncountable cardinal $$\kappa$$?

I've been unable to make progress in determining the answer. Any help would be appreciated!

This theory is complete, but not $$\kappa$$-categorical. The conditions don't say anything about the number of infinite equivalence classes, so you can construct two models, one with a single infinite equivalence class, the other with $$\kappa$$-many such classes. These are clearly not isomorphic, so the theory is not categorical.
In fact, for every infinite cardinal $$\kappa$$ and every model M of cardinality $$\leq \kappa$$ we can find an elementary extension $$N_\kappa$$ of $$M$$ with $$\kappa$$ infinite equivalence classes and with each infinite equivalence class having cardinality $$\kappa$$. This extension is unique up to isomorphism for each cardinal $$\kappa$$, so the theory is complete.
• The latter. If you had a model in which $\varphi$ holds, and one in which $\neg \varphi$ holds, both of cardinality $\kappa$, then one of them would not admit $N_\kappa$ as an elementary extension. – Z. A. K. Feb 4 at 5:42
• Got it. You've already answered my original question, but if you'd be willing to sketch how we construct $N_{\kappa}$, that would be great. – CuriousKid7 Feb 4 at 6:07
• @Z.A.K. Just a small correction. It's not clear from the construction you give that the resulting model has the property that every equivalence class has cardinality $\kappa$. When you find a model by compactness, it might have new equivalence classes of size smaller than $\kappa$. But if you repeat your construction $\omega$-many times, building an elementary chain, you'll get what you want, since then each equivalence class had to be added at some point in the chain. – Alex Kruckman Feb 4 at 12:55
• I mean that if $M$ is the union of an increasing chain of structures, for any equivalence class in $M$, if $m$ is a representative of that class, then $m$ is an element of some structure $M_i$ in the chain. Based on Z.A.K.'s construction, the equivalence class of $m$ is blown up to size $\kappa$ in $M_{i+1}$, so it has size $\kappa$ in $M$. – Alex Kruckman Feb 4 at 18:00