- #1

disregardthat

Science Advisor

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[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?