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edited Feb 7, 2018 at 20:17

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Unless you are committed to beginning with MK class theory, which is not conservative over ZFC, I suspect you can get all you want working in Ackermann's set theory as developed by W. Reinhardt in Ackermann's set theory equals ZF, Ann of Math Log 2, pp. 189-249. There is a nice overview of the theory by Azriel Levy in The Role of Classes in Set Theory, which appears both as chapter of Foundations of Set Theory (Second Revised Edition), A. Fraenkel, Y. Bar-Hillel and A. Levy, North-Holland Publishing Co. (1973) and as a chapter of Sets and Classes (G.H. Müller ed), North-Holland Publishing Co. (1976).

Edit. I wrote this before I saw Joel's comment.

In Reinhardt's version of Ackermann's theory, which is conservative over ZFC (as well as over NBG with Global Choice), given a class $A$ having the power of $On$ one can form $P(A), PP(A), PPP(A), ...$, where $P(A)$ is the power class of $A$.

For some reason Ackermann's theory has not received much attention. Perhaps Joel or someone else knowledgable about such matters can explain why this has been the case.

Edit. I wrote this before I saw Joel's comment.

Unless you are committed to beginning with MK class theory, which is not conservative over ZFC, I suspect you can get all you want working in Ackermann's set theory as developed by W. Reinhardt in Ackermann's set theory equals ZF, Ann of Math Log 2, pp. 189-249. There is a nice overview of the theory by Azriel Levy in The Role of Classes in Set Theory, which appears both as chapter of Foundations of Set Theory (Second Revised Edition), A. Fraenkel, Y. Bar-Hillel and A. Levy, North-Holland Publishing Co. (1973) and as a chapter of Sets and Classes (G.H. Müller ed), North-Holland Publishing Co. (1976).

Edit. I wrote this before I saw Joel's comment.

In Reinhardt's version of Ackermann's theory, which is conservative over ZFC (as well as over NBG with Global Choice), given a class $A$ having the power of $On$ one can form $P(A), PP(A), PPP(A), ...$, where $P(A)$ is the power class of $A$.

For some reason Ackermann's theory has not received much attention. Perhaps Joel or someone else knowledgable about such matters can explain why this has been the case.

Unless you are committed to beginning with MK class theory, which is not conservative over ZFC, I suspect you can get all you want working in Ackermann's set theory as developed by W. Reinhardt in Ackermann's set theory equals ZF, Ann of Math Log 2, pp. 189-249. There is a nice overview of the theory by Azriel Levy in The Role of Classes in Set Theory, which appears both as chapter of Foundations of Set Theory (Second Revised Edition), A. Fraenkel, Y. Bar-Hillel and A. Levy, North-Holland Publishing Co. (1973) and as a chapter of Sets and Classes (G.H. Müller ed), North-Holland Publishing Co. (1976).

In Reinhardt's version of Ackermann's theory, which is conservative over ZFC (as well as over NBG with Global Choice), given a class $A$ having the power of $On$ one can form $P(A), PP(A), PPP(A), ...$, where $P(A)$ is the power class of $A$.

For some reason Ackermann's theory has not received much attention. Perhaps Joel or someone else knowledgable about such matters can explain why this has been the case.

Edit. I wrote this before I saw Joel's comment.

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created Feb 7, 2018 at 20:13

6.5k
1
42
37

Unless you are committed to beginning with MK class theory, which is not conservative over ZFC, I suspect you can get all you want working in Ackermann's set theory as developed by W. Reinhardt in Ackermann's set theory equals ZF, Ann of Math Log 2, pp. 189-249. There is a nice overview of the theory by Azriel Levy in The Role of Classes in Set Theory, which appears both as chapter of Foundations of Set Theory (Second Revised Edition), A. Fraenkel, Y. Bar-Hillel and A. Levy, North-Holland Publishing Co. (1973) and as a chapter of Sets and Classes (G.H. Müller ed), North-Holland Publishing Co. (1976).

Edit. I wrote this before I saw Joel's comment.

In Reinhardt's version of Ackermann's theory, which is conservative over ZFC (as well as over NBG with Global Choice), given a class $A$ having the power of $On$ one can form $P(A), PP(A), PPP(A), ...$, where $P(A)$ is the power class of $A$.

For some reason Ackermann's theory has not received much attention. Perhaps Joel or someone else knowledgable about such matters can explain why this has been the case.