Re: Well Ordering the Reals


David R Tribble wrote:
> Randy Poe wrote:
> >> If you want to talk about complements, then you need a "universe".
> >> If you don't care about complements, you don't need a
> >> "universe" as far as I know.
> >
>
> Albrecht Storz wrote:
> >> I think you should care about complements. How would you surely prevent
> >> the build up of complements accidentially?
> >
>
> Virgil wrote:
> > By restricting construction to relative compliments: the compliment of
> > set A relative to set B is the set of elements in B but not in A.
> > These are also called difference sets, frequently denoted by 'B\A'.
> >
> > Given any two sets, one can construct from them 4 possibly different
> > sets: their union, their intersection and their two differences.
> >
> > Given any family of sets, the nearest one need get to a universal set is
> > their union, as all the sets constructable by union, intersection and
> > differencing from members of that family will be subsets of that union.
>
> Perhaps Albrecht is trying to make a point about notS being impossible
> using some other construction from S than "complement".
>
> Consider universe U containing all the naturals and all the possible
> sets of naturals. Is U a set or a class?
>
> Now consider E, the set of all even naturals from U. Obviously
> the set E exists as an object in universe U, as do all the member
> naturals in E.
> E = {0, 2, 4, 6, 8, ...}
>
> Now consider notE to be the complement of set E, i.e., the set
> having as members all the naturals in U that are not also
> members of E. So notE is the set of all odd naturals.
> notE = {1, 3, 5, 7, 9, ...}
>
> However, I think perhaps Albrecht considers notE to be more than
> just the natural objects from U, i.e., notE is the set of all _objects_
> in U that are not members of E. This includes sets as well as
> naturals, so notE is the set of all odd naturals, plus all the
> possible subsets of the naturals (which are not members of E).
> notE = U \ E
> = {1, 3, 5, 7, ...}
> u {{0}} u {{1}} u ...
> u {{0,1}} u {{0,2}} u ...
> u ... u {{0,1,2,3,...}} u ...
>
> Does this lead to a contradiction? Is notE still a proper set?


If U and E are proper sets, yes.
The naturals and the elements of P(N) are axiomaticly created objects
in ZFC. By definition their "elementness" is guaranteed. All sets which
are build up out of this universe exist.

It's so clear because your "univers" is a basic one of ZFC.
Look at another kind. Let's ask: Exists the set of all lines inclusive
the straight line in ZFC?
Now at first, you have to build up lines out of the elements of ZFC.
How do you do this with the basic objects of ZFC?


Regards
Albrecht Storz

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >> I think you should care about complements. ... > differencing from members of that family will be subsets of that union. ... Perhaps Albrecht is trying to make a point about notS being impossible ... Consider universe U containing all the naturals and all the possible ...
    (sci.math)
  • Re: Orlow cardinality question
    ... >>> members in an arbitrary finite set. ... >> I have talked about the two unit infinities ... >>> naturals, it says nothing about the set itself. ... which is the standard set for finite cardinality. ...
    (sci.math)
  • Re: infinity
    ... > If this is the case, which members of O are not members of O*? ... > borrowing, the second I call it definition with borrowing, It is very ... > condition) from another infinite set Q which is bigger than the ... but are not in the other infinite set of odd naturals. ...
    (sci.math)
  • Re: Orlow cardinality question
    ... A set is a number of elements, members, units, or whatever. ... > some member of the sequence) of, for decimal numbers, either the ... I have talked about the two unit infinities and the ability to classify all ... >> Yeah like the entire set of naturals. ...
    (sci.math)
  • Re: Orlow cardinality question
    ... >> using some properties of the members to form correspondences between ... > Then measure the relative "density" of the set of naturals and the set ... > Apply this theory to the set of permutations on the set of naturals. ... is the set of permutations defined using a linear mapping function on N? ...
    (sci.math)