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Thread: Sets containing unions, and vice versa

  1. #1
    May 2011

    Sets containing unions, and vice versa

    I've been going through some old exam papers, and I came upon an interesting part of a question that I'm not sure how to solve. Throughout the question I've been proving things like $\displaystyle b\in b$ is false, at most one of $\displaystyle b\in c$ and $\displaystyle c\in b$ is true, and so on. I've been asked whether any of the following can occur, using only axioms (ZF1)-(ZF4) and (ZF9). (Extensionality, null set, pairs, unions, foundation.)

    1. $\displaystyle x\in\bigcup x$
    2. $\displaystyle x\in\bigcup \bigcup x$
    3. $\displaystyle \bigcup x \in x$

    I think (1) is false, by extension I think (2) is false (but can't get my reasoning right in my head), and I'm not sure about (3).
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  2. #2
    Senior Member
    Feb 2010
    3. U{0} e {0}.

    Indeed, for any ordinal x, we have U(x+) = x and x e x+.
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