Re: Cantor's definition of set

On Sat, 27 Oct 2007 14:32:47 +0200, Jan Burse <janburse@xxxxxxxxxxx>
wrote:


Sets are nameless.

No, not really. (Depends on the very system considered.)

Actually, set abstracts ( say "{x : x =/= x}" ) are terms, i.e. names
(again depending on the very system considered, of course).


This follows from the axiom of "extensionality".
This axiom exactly says that sets do not have names.

But wait a second... if I get you right, then not even people "have"
names. After all a name is not some mark on the foreheads of people...

But names are USED to refer to certain people. In the same way we use
certain names (for example set abstracts) to refer to (certain) sets.

For example, in ZFC we may use the name "{x : x =/= x}" to refer to the
empty set.


Because the axiom says that sets are already equivalent when they
have the same elements in it.

No. The axioms doesn't say that they are "equivalent", but IDENTICAL
- if we formulate our set theory in a framework of FOPL _with identity_.

A = B

HERE means that A and B are _identical_; not just "equivalent".


So there is no room for a name.

Of course there is. Actually, there are variants where the symbol "0" is
a primitive, and we have an axiom

~Ex(x e 0).

If we read "~Ex(x e y)" to mean: y is empty. The axiom states that 0 is
empty. With other words (given extensionality) 0 is _the_ empty set. And
this in turn means: "0" denotes the empty set in this theory.

No room for a name? Huh?!


F.

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