# Is ZFC + classes equi-interpretable with MK?

If we add to the language of ZFC class symbols, denoted using letter style $$\mathcal X, \mathcal Q, \mathcal R, \mathcal P, \mathcal L,.."$$, call it class style, then add the axiom schema of class comprehension: $$\exists \mathcal X \forall \mathcal Y \ [\mathcal Y \in \mathcal X \leftrightarrow \exists y=\mathcal Y :\phi(y)]$$, Where $$\phi(y)$$ is any formula in which $$y$$ only occur free and in which $$\mathcal X$$ doesn't occur.

Add all axioms of ZFC written in non-class style.

Add the axiom of sorting: $$\forall x \exists \mathcal Y (x= \mathcal Y)$$ Add the axiom of membership: $$\forall x \forall \mathcal Y (\mathcal Y \in x \to \exists y (y=\mathcal Y))$$

Call the resulting theory "ZFC + classes".

Notice that ZFC + classes is different from NBG\MK where every member of a class is a set, here only members of sets are sets, so you can have proper classes (i.e. non-sets) that are members of proper classes.

Is ZFC + classes equi-interpretable with MK?

If instead of adding ZFC axioms in non-class style only, we add all homogeneously written axioms of ZFC.

Where in a homogeneous expression either all terms are written in non-class style or all terms are written in class style.

Call the resulting theory ZFC+$$classes_2$$

Is ZFC+$$classes_2$$ consistent?

If consistent, then what's its consistency strength?

To be noticed is that if ZFC+$$classes_2$$ is not stronger than ZFC, then it would be a candidate theory for foundation of mathematics according to Muller's criteria!

• can a similar argument prove the same result for ZFC + $classes_2$? Something similar to $V_\alpha \prec V$ – Zuhair May 7 '20 at 21:13