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2-By A'2, every set y satisfying axiom A' must be a transitive set. But it is not true that every set y satisfying axiom A must be transitive. So, it seems natural to ask the following. Question 2: (i) If a set y satisfies axiom A, is it necessary that the transitive closure of y is a set satisfying axiom A? (ii) If x is a transitive set, does any set y satisfying the axiom A and having x as a member set be a transitive set ?

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@Gerard: I see that you are asking a whole lot of questions about the so-called axioms A and A'. While it is good form to split separate questions into separate posts, it may also be good form to either (1) explain what axioms A and A' are in every post (as it stands this question doesn't mean anything by itself) or (2) link to your previous post that states the two axioms. –  Willie Wong Jun 30 '10 at 11:43
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You are perfectly right, but I am a beginner with mathoverflow and do absolutely not know how to do this ! Ideally, I would have written only one message with the whole stuff, but it seems that this is too big, so i felt obliged to write five distinct linked messages. Gérard LANG –  Gérard Lang Jun 30 '10 at 12:37
    
The problem is that the messages won't be automatically linked like that. Try copying the url from the first post (which Ben Webster cleaned up for you), and at the very least edit all the subsequent posts to contain that url somewhere, so that the definitions can be easily found. For an enumerated list, try also formatting it using the function built in to the editor on MO. –  Willie Wong Jun 30 '10 at 13:00

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