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> You say, "Set theory deals with the properties of sets, not elements of
> sets. The only
> property that elements have is set membership."
>
> I understand this, but I see a poosibility for an "element" to have a
> property like "membership in any set is invalid". That is why I think
> the a formal definition of "element" should be given before we can talk
> about any membership.
Not "invalid". You would form an additional Axiom like:
The exists v such that for all x, v is not an element of x.
(Ie this axiom creates an element that is not a member of any set).
This is probably OK as long as v is not a set. In most versions of set
theory, sets can only be constructed from other sets, so v wouldn't really
get involved.
However, if v can be a set, you are stuffed. By the Axiom of the Power Set,
v is an element of its own power set, and so v is an element of a set. This
means your new Axiom is inconsistent with the rest of set theory. It would
be like trying to introduce a new axiom into arithmetic that 1=2.
> And, BTW, I don't know what ZFC is. Is it someting important to know?
> Or all axioms would be still the same?
>
A version of ZF with an additional Axiom (called the Axiom of Choice, hence
the "C"). The difference doesn't matter here I suspect.
http://en.wikipedia.org/wiki/Zfc
Peter Webb
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