Cardinals definable using ordinal arithmetics Let $\kappa$ be an infinite cardinal.
$\kappa$ is stable under ordinal addition $+$, ordinal multiplication $.$ and ordinal exponentiation $e: (a,b) \mapsto a^b$, so $\mathcal{K} = (\kappa,+,.,e,\in)$ is a model theoritic structure.
a) Is the set of cardinals $< \kappa$ first order definable in $\mathcal{K}$?
b) Which cardinals $< \kappa$ are first order definable in $\mathcal{K}$?
Note that if $\{|\lambda| \ | \ \lambda \in \kappa\}$ is definable then so are all $\aleph_{\alpha}, \alpha < 2.\omega_0$ (provided they are in $\lambda$), and you can probably go much further. 
Not too much however, for one can only define countably many cardinals, so there is a least undefinable cardinal $\leq \aleph_{\omega_1}$.
 A: a) Note that $\mathcal K =(\kappa, +, e, \in)$ is absolute between transitive models of ZFC. Thus any collapse below $\kappa$ demonstrates that the set of cardinals below $\kappa$ cannot be first order definable in $\mathcal K$ if $\kappa > \omega_1$.
b) Sorry, I'm not sure how to prove my initial claim that the answer is $\omega + 1$. I will add to this answer in a moment.

Actually, the situation is easier: If $\kappa$ is uncountable, collapse $\kappa$ to $\omega_1$. As $(\omega_1,+,e,\in)$ cannot define uncountable cardinals, absoluteness yields the result.

To clearify: What this answer actually says: Let $\phi(x)$ be a $\{+,e,\in\}$-formula and let $\kappa$ be an ordinal. We cannot prove (in ZFC) that
$$\phi( (\kappa,+,e,\in) ) := \{ \alpha < \kappa \mid (\kappa,+,e,\in) \models \phi(\alpha) \}
$$
contains a cardinal above $\omega$.
On the other hand, let 
$$
\phi(x) = x \in  \omega \vee x = \omega \vee \forall y \colon y = x \vee y \in x
$$
Then $\phi((\omega_1+1; \in)) = \omega+1 \cup \{\omega_1\}$ and thus $\phi(x)$ defines a set of cardinals, that contains an uncountable cardinal. This may seem very strange to you, but what my answer basically comes down to: Above $\omega$, ZFC tells us nothing about which ordinals are cardinals... (Even if we restrict our attention to transitive models with "the true" membership relation.)
