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The set of all sets contradiction

  1. Feb 12, 2009 #1
    1. The problem statement, all variables and given/known data

    Let A be the set of all sets.

    #1.) Show that P(A) is a subset of A.

    #2.) Find a contradiction.

    2. Relevant equations

    3. The attempt at a solution

    #1.) We know that P(A) is a set. Therefore it must be in A, since A is the set of ALL sets.

    #2.) I cannot figure out what to do here. I do not have a theorem that says that |P(A)| > |A|. I do have a theorem that states that there is no injection from P(A) to A and also that there is no surjection from A to P(A).

    Since there is no injection from P(A) to A, does this alone mean that |P(A)| > |A| ?
  2. jcsd
  3. Feb 12, 2009 #2


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    Yes, it does. If |C|<=|D| then there is an injection from C->D. If there is no injection then the opposite must hold so |C|>|D|.
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