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Math Help - ZF Axiom of regularity

  1. #1
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    ZF Axiom of regularity

    I am a little confused over the axiom. Does it mean we cannot have set such that: A = {B, C} and B = {C, D}

    According to Venn diagram, this seems trivial but is against the foundation axiom ∀A≠∅: ∃B∈A: A∩B≠∅

    I mean how can a member of set be disjoint from the set ??
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  2. #2
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    Hi
    The axiom of regularity just states that a set has to contain an element which has no common element with the set.

    Does it mean we cannot have set such that: A = {B, C} and B = {C, D}
    No, sets of this form exist: if C does not contain (as elements) B nor C, then C\cap A=\emptyset and there is no problem.

    A consequence of this axiom is that for any set x, x\notin x, that x_1\in x_2\in ...\in x_n\in x_1 is impossible for any n sets x_i and more generally, there can't be any infinite downward \in-sequence between elements of a set.

    But this axiom does not belong to the theory Z, and I've heard that some researchers are exploring set theories with its negation.
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