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Math Help - Axiom of Separation Problem

  1. #1
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    Axiom of Separation Problem

    Let \phi (v) be a formula of the languages of set theory. Let X be a set. Then the following is a set: \{a \in X | \phi [v] \}, ie all the elements of X with the property \phi form a set.

    Write down a formula \phi (v) saying that the set v has exactly one element.

    I'm actually stuck here for very long. Is there really an abstract formula that represents all the concrete examples that the set has exactly one element?
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  2. #2
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    Re: Axiom of Separation Problem

    Quote Originally Posted by Markeur View Post
    Write down a formula \phi (v) saying that the set v has exactly one element.
    [I'll use 'E' for 'there exists', 'e' for 'is an element of' and 'A' for 'for all']

    Ex(xev & Ay(yev -> y=x))

    By the way, you probably meant not

    {aeX | P[v]}

    but rather

    {veX | P[v]}
    where the variable 'X' does not occur free in P[v]

    or

    {aeX \ P[a]}
    where the variable 'X' does not occur free in P[v]
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