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Self referance

  1. Sep 16, 2009 #1


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    Russels paradox says that the set of all sets not contained in themselves, i.e.

    [tex]x=\{ y \ : \ y \not \in y \}[/tex]

    neither is or is not contained in itself. The set can be created based on an axiom saying that
    "The set of objects with a property Q exists".

    Let`s assume Q : "Is not contained within itself" is a valid property. Then not Q also is a valid property. Assume z has the property not Q. Then

    [tex]z \in z[/tex]

    What we do when defining such a set is:

    [tex]z=\{ y \ : \ y \ \text{has} \ P \} \cup \{ z \}[/tex] where P is some property.

    Is this in any sense a valid definition? Before z is properly defined, we use it. Any definition must consist of properly defined terms, and in this case i argue it doesn`t. Hence not Q is not a valid property, and thus Q isn`t.

    ZFC fixes this by saying that a set can be created as long as it isn`t equivalent with Russels paradox. Can`t we just say that the property must be well defined?
  2. jcsd
  3. Sep 16, 2009 #2
    Russel's paradox says that the existence of a set [tex]
    x=\{ y \ : \ y \not \in y \}
    leads to a contradiction. It was for this reason why ZF was created, which has the axiom of regularity, preventing the question of self-inclusion all together.

    If we're working in ZF, then your question about [tex]
    z=\{ y \ : \ y \ \text{has} \ P \} \cup \{ z \}
    [/tex] is uninteresting. The set z contains itself, and so within ZF, it does not exist.

    There are other kinds of self-reference which do not create any sort of contradiction. It's possible in ZF to define functions, sequences, and other nice things recursively.
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