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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!

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ZFC+π‘π‘™π‘Žπ‘ π‘ π‘’π‘ 2 is equiconsistent with ZFC. Let LZFC be the theory obtained from ZFC by adding a constant symbol 𝛼, an axiom asserting that 𝛼 is an ordinal, and for every formula F of ZF with parameters from 𝑉𝛼 an axiom asserting that F holds in 𝑉𝛼 iff F holds. By reflection one can see that LZFC is a conservative extension of ZFC. If we interpret set as an element of 𝑉𝛼, and everything as a class, then all the axioms of ZFC+π‘π‘™π‘Žπ‘ π‘ π‘’π‘ 2 hold.

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  • $\begingroup$ can a similar argument prove the same result for ZFC + $classes_2$? Something similar to $V_\alpha \prec V $ $\endgroup$ – Zuhair May 7 '20 at 21:13
  • $\begingroup$ @Zuhair: Yes and I have edited the answer to indicate this. $\endgroup$ – Greg Kirmayer May 7 '20 at 23:28
  • $\begingroup$ when you said "by reflection", which reflection theorem of ZFC you are using? $\endgroup$ – Zuhair May 8 '20 at 9:01

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