Re: The empty set

On Dec 14, 3:49 am, G. Frege <nomail@invalid> wrote:
On Thu, 13 Dec 2007 14:08:19 -0800 (PST), apoorv <sudhir...@xxxxxxxxxxx>
wrote:



The empty set is
0 = {x : x e A and ~x e A}.

So, For all x, x e 0 <--> x e A and ~x e A,
or, x e 0 <--> ~[x e A or ~x e A]
or. x e 0 <--> ~[x e A u Ac] , where Ac is the complement of A ,

No - at least not in the context of our "standard" set theory ZF(C),
since there's no "absolute" complement, only a /relative complement/.

So -for the sake of the argument- let's consider a set theory, where

~(x e A) <-> x e A'

(assuming that there is such a theory).



or, x e 0 <--> ~[x e U], where U is the domain for the variable x.

No. Rather, where U is the /universal set/ - a set containing all
objects of our theory.



So, the empty set is the complement of the domain U of the variable x.

The empty set is the (absolute) complement of the universal set: the
contrary of /everything/ is /nothing/.



Suppose the domain U is contained in a larger domain U'.

IF U is the universal set, THEN there is no larger set.

Note that in ZF(C) there is no universal set.
If there is no universal set (containing all objects of the theory),
then
how can we have 0 as the absolute complement of the non existing
'universal set'?
If we take our universe as the class of all sets, then would 0 be the
class of all
'non sets'?There is at least one non set, namely, the class of all
sets.
-apoorv


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