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Let $SC$ (sets and classes) denote the modification of Ackermann set theory laid out in F. A. Muller's Sets, Classes and Categories (denoted there by ARC -- a brief summary of the theory is given below, see the reference for a more detailed exposition).

Muller's paper lays out how we can handle all of standard $1$-category theory in this modified theory (summarized below), and further points out that $SC$ is not conservative over $ZFC$ but that $ZFC^+$ ($ZFC$ plus the existence of the first non-denumerable strictly inaccessible cardinal) is sufficient to prove its consistency, whereas MacLane's set theory requires $ZFC^{++}$ to establish consistency and Grothendieck's theory with universes requires $ZFC^{++\cdots}$.

Let $SC$ (sets and classes) denote the modification of Ackermann set theory laid out in F. A. Muller's Sets, Classes and Categories (denoted there by ARC -- a brief summary of the theory is given below, see the reference for a more detailed discussion).

Muller's paper lays out how we can handle all of standard $1$-category theory in this modified theory (summarized below), and further points out that $SC$ is not conservative over $ZFC$ but that $ZFC^+$ ($ZFC$ plus the existence of the first uncountable strictly inaccessible cardinal) is sufficient to prove its consistency, whereas MacLane's set theory requires $ZFC^{++}$ to establish consistency and Grothendieck's theory with universes requires $ZFC^{++\cdots}$.

1. Extensionality, in the usual sense applied to classes.
2. Completeness, asserting that $\mathbb{V}$ is closed under members of members and subsets of members.
3. Class Separation, asserting that for any predicate $\phi(\cdot,Y)$ where $Y$ stands for any finite number of class parameters, and for every class $Z$, there exists a class $A$ whose members are exactly those members of $Z$ satisfying $\phi(\cdot,Y)$.
4. Regularity, but only for sets.
5. Choice, again for sets only.

1. Extensionality, in the usual sense applied to classes.
2. Completeness, asserting that $\mathbb{V}$ is closed under member’s members and member’s subsets.
3. Class Separation, asserting that for any predicate $\phi(\cdot,Y)$ where $Y$ stands for any finite number of class parameters, and for every class $Z$, there exists a class $A$ whose members are exactly those members of $Z$ satisfying $\phi(\cdot,Y)$.
4. Regularity, but only for sets.
5. Choice, again for sets only.

Muller's paper lays out how we can handle all of standard $1$-category theory in this modified theory (summarized below), and further points out that $ZFC^+$ ($ZFC$ plus the existence of the first non-denumerable strictly inaccessible cardinal) is sufficient to prove its consistency, whereas MacLane's set theory requires $ZFC^{++}$ to establish consistency and Grothendieck's theory with universes requires $ZFC^{++\cdots}$.

Muller's paper lays out how we can handle all of standard $1$-category theory in this modified theory (summarized below), and further points out that $SC$ is not conservative over $ZFC$ but that $ZFC^+$ ($ZFC$ plus the existence of the first non-denumerable strictly inaccessible cardinal) is sufficient to prove its consistency, whereas MacLane's set theory requires $ZFC^{++}$ to establish consistency and Grothendieck's theory with universes requires $ZFC^{++\cdots}$.