# Reference request: Reflection as a single axiom interpreting the whole of ZFC?

Working in first order languge $$\mathcal L(\in, W)$$, where $$W$$ is a constant symbol.

Reflection: if $$\varphi$$ is a formula in $$\mathcal L(\in)$$, in which $$x$$ is free, and $$\vec{p}$$ is the string of all of its parameters, and if $$\psi$$ is a formula in which $$z$$ is free and $$y$$ not free, then all closures of: $$\vec{p} \in W \wedge \exists x (\varphi) \to \exists x [\varphi\wedge \exists y \in W \ \forall z (z \in y \leftrightarrow z \in x \wedge \psi) ]$$; are axioms.

Now with Extensionality this would interpret all axioms of $$\text{ZFC}$$ since it would trivially interpret Harvey Friedman $$\text{K(W)}$$ theory (page 3). Although I'm not sure but I think even Extensionality is interpretable using the above scheme only.

Has there been prior attempts to non-trivially shortly axiomatize ZFC with a single comprehension schema? and had this schema been studied before specially in absence of Extensionality?

• Well, the Replacement scheme implies the Separation scheme, so the answer to your first question is immediately "yes." – Noah Schweber Jan 25 at 18:51
• To me all of the axioms of set union, power and infinity, all are individual comprehension axioms. What is not a comprehension axiom is Extensionality, Foundation and Choice. And by the way Replacement doesn't interpret all the others – Zuhair Jan 25 at 19:10