Compactness for definable models?

Question

In this question, I use "model" to mean a model in the language $\{\in\}$ of set theory. Call a model $M$ "definable" iff for every $x \in M$, there is a formula $\phi(\vec{y},z)$, where $\vec{y} \in M$ is a sequence of parameters, such that $M \models \forall z (z \in x \leftrightarrow \phi(\vec{y},z))$.

I want the following variation on the compactness theorem: given a theory $T$ (in the language $\{\in\}$), if every finite subset of $T$ has a definable model, then $T$ has a definable model. Is this true?

[Edit: the question above is not interesting, as explained in an answer below. A potentially interesting variant is as follows. Say that a model $M$ is "definable" iff for every $x \in M$ there is a formula $\phi(z)$, without parameters, such that $M \models \forall z (z \in x \leftrightarrow \phi(z))$. Ask the same question. Or, ask the same question using pointwise definability, as described by Andres Caicedo.]

Thank you!

Answer 1

The question is not an interesting one, as Arthur Fischer explains:

This appears to be a very weak form of definability. In particular, it seems that every $\{∈\}$-model is definable. Note that for $x∈M$ we have that $M⊨(∀z)(z∈x↔z∈x)$, and I cannot see any reason $x$ cannot be used as a parameter for itself. In the talk Andres Caicedo linked to, Joel Hamkins uses a much stronger form of definability: $a∈M$ is definable iff there is a formula $ϕ(x)$ such that $M⊨ϕ[b]$ iff $b=a$ for all $b∈M$. – Arthur Fischer

Additionally, Andres Caicedo adds relevant information:

Nick, there are pointwise definable models of set theory. You may want to look at this link for some context, and a paper: http://jdh.hamkins.org/math-tea-argument-vienna-2011 Anyway, here is a sketch: If there is a model of ZFC, then there is one, call it M , satisfying V = L. This model admits definable Skolem functions, so we can form the Skolem hull H of the empty set. This is an elementary substructure of M , thus a model of ZFC, and it is pointwise definable. Note that it suffices that M satisfies V = HOD and that, if M is transitive, then the transitive collapse of H exists. – Andres Caicedo