Question related to "Foundations of Mathematics"

[mv_Cristi]

I made up it, I didn't find anything about it, so I need help.

The question is: How do different set theories handle the following
definition?

U := {x : (x belongs_to A) and ( Not(A is_included_in Powerset(
EmptySet)) )}

If possible in the set theory, require (A is_set). But it doesn't
really matter.

The set U is interesting for some reasons.

The answer will help me a lot in my PhD. research.

Thank You.

[smn]

Hello, when you say A is _included_ in ... , do you mean A is a
subset of, which would mean here that either A=empty set or A= the set
whose only member is the empty set , or do you mean A is an element of
(or belongs to) which would make A= the empty set .
Assume you mean "subset" ,then if A is not the empty set and not
the set whose only member is the empty set then U=A ,otherwise U= the
empty set . We can set f(A)=U "f(A)" is called a defined term in one
free variable which associates to each set A a new set f(A) ,"f" is
often called a function symbol an then informally said to denote an
operation or function on sets but if so this function is not an object
of set theory ,i.e f is not a set. The predicate "f is a function" can
be defined in the set theory (as a SET of ordered pairs ,no two of
which have the same first element) where all the variables must denote
sets .But this particular f discussed here is not a set .
If you allow proper classes ( objects which are not elements of
other objects as in Kelly -Moore theory and if you restrict your A
here to be a set (i.e. objects which are elements (members) of other
objects then f is a function ,but not a set ,it is a proper class .
Regards,smn

[smn]

On Mar 10, 10:20 am, mv_Cristi <mv_cri...@yahoo.com> wrote:

Hello , U = A unless A is either emp (the empty set) or the set {empt}
whose only element is emp in which case U=emp . What do you wish to
know ? smn