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I'm reading a book about Zermelo–Fraenkel set theory, and I was told that there are a lot of dependence relation among the commonly used axiom system (in the question What axioms does ZF have, exactly? , many examples can be found). But I notice that most of dependence relations are cause by the Replacement Scheme. So I wonder that if only considering the axioms of Zermelo set theory (that is, ZC, or more precisely, the axiom system consisting of Extensionality, Foundation, Comprehension Scheme, Pairing, Union, Infinity, Power Set and Choice), does there still remain any redundant axiom? Thanks in advance.

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Censi LI
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    Some axioms in the comprehension schema imply each other because the formulas involved are provably equivalent. – Trevor Wilson Apr 16 '15 at 08:12
  • All the models exhibiting independence in the case of $\sf ZFC$ can be used in this case. So what if in some of these models Replacement is true? – Asaf Karagila Apr 16 '15 at 09:33
  • @Asaf: I think the idea of the question was that many of the axioms are equivalent in the presence of replacement, so we might be able to get more independence by looking at models that don't satisfy replacement. You are pointing out that the ones that are already independent over ZFC are also independent over Z. – Carl Mummert Apr 16 '15 at 11:17
  • @Carl: Well, in some sense, most of them are independent (I'm treating the schema as a single axiom here). – Asaf Karagila Apr 16 '15 at 11:19
  • I wonder whether any of this is in the old Foundations of Set Theory book by Fraenkel, Bar-Hillel, and Levy. Independence of axioms sounds like the type of thing they may have written about. – Carl Mummert Apr 16 '15 at 11:24
  • @TrevorWilson You are right. And it seems cumbersome to consider for which formula the corresponding Comprehension axiom is independent of other axioms. So let's drop this case... – Censi LI Apr 16 '15 at 18:00
  • @AsafKaragila You've told me that each ofUnion, Infinity and Power Set are independent from the others. And it's a celebrated result that Choice is independent from the others, too. Do you know any other obvious dependence or independence relation among them? Thanks in advance. – Censi LI Apr 16 '15 at 18:03

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