In this article, the author, F.A.Muller, suggests criteria for a founding theory of mathematics (pp:14-16). The author proposes $ARC$ Class Theory to embody these requirements. The motivation is largely connected to set theoretic basis for category theory.

The first Question: how far the following theory $``\mathsf{RfST}"$, is from fulfilling these requirements?

I mean the *Class Separation schema* he speaks of seems to be in some sense out of touch with what class theories are about, one needs in my opinion to speak of a more general theme that explains it. In other words it needs to be a theorem schema of some more basic principle that rise in connection with a theory about Classes and Sets. Hence the following theory suggested. The point is that the axiomatic system here is more elegant, and I think it serves the purpose.

However I'm not sure if the *Counter-Reflection* principle is too strong for the purposes outlined in Muller's article, specifically regarding adhering to Bourbaki without *Super-abundancy*, since I tend to think that Counter-Reflection doesn't add strength. Hence the question.

The next question: how much the axioms of $\mathsf{RfST}$ are far from Harvey Friedman's criteria for new axioms for mathematics, written in this article (see 5. CIRCUMSTANCES SURROUNDING ACTUAL ADOPTION OF NEW AXIOMS. pp:5-6)?

Reflective Set Theory $\mathsf{RfST}$ is formulated in mono-sorted first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ denoting the class of all sets.

The axioms are those of first order identity theory +

**Extensionality:**$\forall x (x \in a \leftrightarrow x \in b) \to a=b$**Class comprehension:**if $\varphi(y)$ is a formula in which the symbol $``y"$ occurs free, then all closures of: $\exists x \forall y (y \in x \leftrightarrow y \in V \wedge \varphi(y))$, are axioms.**Super-Transitivity:**$ y \subset x \wedge x \in V \to y \in V$**Reflection:**if $\varphi(y, x_1,..,x_n)$ is a formula that doesn't use the symbol $V$, in which only $y,x_1,..,x_n$ occur free, then:

$$\forall x_1,..,x_n \in V \\ [\exists y (\varphi(y,x_1,..,x_n)) \to \exists y \in V (\varphi(y,x_1,..,x_n))]$$

$\hspace{12pt}$ is an axiom.

**Counter-Reflection:**if $\varphi$ is a sentence that doesn't use the symbol $V$, and $\varphi^V$ is the sentence obtained by merely bounding every quantifier in $\varphi$ by $V$, then: $ (\varphi^V \to \varphi) $ is an axiom.**Foundation:**$\exists m\in x \to \exists y \in x \forall z \in x (z \not \in y)$**Choice:**$\forall m,n \in X ( \not \exists k \in m (k \in n)) \to \\\exists Y \forall x \in X (x \neq \emptyset \to \exists! y \in Y (y \in x ))$