Can there exist a world where every consistent effective FOL theory can be written and satisfied in some part of it?

Question

is it consistent to postulate the existence of a set $W$ and a membership relation $\in$ and the following axioms about it:

1. Transitivity: $\forall x \in W: x \subset W$

2. Extensionality: $\forall x,y \in W: \forall z (z \in x \iff z \in y) \to x=y $

3. Comprehension: $\exists x \forall y (y \in x \iff y \in W \land \phi)$

4. Pairing: $\forall a,b \in W: \{a,b\} \in W$

5. Syntax: Any consistent recursively axiomatized first order theory that doesn't contain $W, \in$ among its symbols, can be written in $W$

6. Semantics: For every consistent recursively axiomatized first order theory $T$ that doesn't contain $W,\in$ among its symbols; there is a subset $M$ of $W$ such that $M \models T$.

In nustshall $W$ is a world in which every consistent effectively generated first order theory can be written and satisfied in a sector of it.

Is this consistent?