# Show that $\Gamma$ is $\kappa$-categorical for $\kappa>\aleph_{0}$ , but not $\aleph_{0}$-categorical.

Let $\mathcal L=\{c_{i}:i<\omega\}$ be a language in first order logic, and:

$\Gamma =\{\forall x(x=x),\forall x\forall y(x=y\rightarrow y=x),\forall x\forall y\forall z(x=y\wedge y=z\rightarrow z=y)\}\cup\{c_{i}\neq c_{j}:i,j\in\omega ,i\neq j\}$

Show that $\Gamma$ is $\kappa$-categorical for $\kappa>\aleph_{0}$ , but not $\aleph_{0}$-categorical.

¿It can be an arbitrary equivalence relation $R$ , not necessarily $=$?

• You shouldn't use "=" for anything other than equality. Please edit to make it clear what the sentences in $\Gamma$ are... – universalset Dec 13 '13 at 3:10
• Hello Yesid.${}$ – Camilo Arosemena-Serrato May 3 '15 at 19:10

Your theory is $\kappa$-categorical for all uncountable $\kappa$, since every model of uncountable size $\kappa$ of the theory has the $\omega$ many distinct constants, plus $\kappa$ many other individuals, and all such models are isomorphic. But your theory is not $\aleph_0$-categorical, since your theory has a model with only the constants, and another with the constants plus another individual that is not a constant, or two others or $n$ others or $\omega$ many others, and these models are not isomorphic. So there are countably many distinct countable models up to isomorphism.
The result isn't true if you interpret $=$ only as an equivalence relation, since in this case, there can be non-isomorphic models of uncountable size $\kappa$, depending on the size of the equivalence class of each $c_i$.