# Outer Definability of a Class

Definition: Let $C$ be a class of sets and $\mathcal{L}$ a first order relational language. We say $C$ is "outer definable" by $\mathcal{L}$ if there is a first order theory $T$ and for each $n_{R}$ - ary relational symbol $R\in \mathcal{L}$ there is a class $E_{R}\subseteq V^{n_{R}}$ such that: $\forall a\in V~~~(a\in C\Longleftrightarrow M_{a}\vDash T)$ which $M_a$ is a $\mathcal{L}$ - structure as follows: $Dom(M_a):=a~~,~~\forall R\in \mathcal{L}~~~~~R^{M_a}:=E_{R}\cap a^{n_R}$

Question 1: Is each class $C$ outer definable in the language of set theory?

If not then:

Question 2: Are $Ord$ and $Card$ outer definable in the language of set theory (or any other suitable relational language)?

Question 3: Is each class $C$ outer definable in some suitable first order relational language?

The answer to question 1 is no. Let $C$ be the class of all countable sets. If the answer were affirmative, we would get the relation $E$ and theroy $T$ so that $a\in C$ if and only if $\langle a,E\upharpoonright a\rangle\models T$. Let $a$ be any uncountable set, so it violates the theory, and now take a countable elementary substructure, which must still violate the theory, but it is countable.
A similar argument shows that Ord and Card are not outer definable in any language, since we may take some enormous ordinal $\alpha$, which must satisfy the theory, but it will have an elementary substructure that is not an ordinal, yet still satisfy the theory.