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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?

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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.

The argument also refutes question 3, if the language is a set.

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  • $\begingroup$ Thank you Prof. Hamkins. What about the non-set languages in the question (3)? Is there any essential difference? How can we deal with languages with class many symbols formally? $\endgroup$
    – user43940
    Commented Dec 10, 2013 at 18:41
  • $\begingroup$ If the language is a proper class, then models are not really sets any more, even when the domain is a set, and the analysis breaks down. One wouldn't expect the Lowenheim-Skolem theorem to provide small elementary substructures in this case, for example. I'm not sure exactly what one could say for proper class languages. $\endgroup$
    – Joel David Hamkins
    Commented Dec 10, 2013 at 22:44

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