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Topic: Given a set , is there a disjoint set with an arbitrary cardinality?
Replies: 28   Last Post: Dec 4, 2012 5:50 PM

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 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Given a set , is there a disjoint set with an arbitrary cardinality?
Posted: Dec 4, 2012 3:39 AM

On Tue, 4 Dec 2012, forbisgaryg@gmail.com wrote:
> On Monday, December 3, 2012 10:29:36 PM UTC-8, William Elliot wrote:
> > On Mon, 3 Dec 2012, forbisgaryg@gmail.com wrote:
> > > On Monday, December 3, 2012 6:21:05 AM UTC-8, jaakov wrote:
> >
> > > > Given a set X and a cardinal k, is there a set Y such that
> > > > card(Y)=k and X is disjoint from Y?

> >
> > > > Is there a proof of this fact that works without the axiom of
> > > > regularity (= axiom of foundation) and does not assume purity of
> > > > sets?

> > > Based upon some of your replies I have a question.
> >
> > > Given your statement: "Given a set X and a cardinal k"
> >
> > > 1. Are you referring to any set X or specifically a set of real
> > > numbers?

> > Any set or course.
> >

> > > 2. Are you saying saying k isn't necessarily the cardinality of X?
> > That's right.
> >
> > All that was said about the given cardinal k was card Y = k.

> So, what is the cardinality of the universal set?

In ZPG and NBG set theory, there no universal set.

> As was noted no set can be a member of itself.

Regularity implies that and the problem asks to not use regularity.

> (Ex ~xeU) <=> ~(Ax xeU)

> Can a subset of the universal set have the same
> cardinality as the universal set?

As noted above, there is no universal set unless you're using
NF set theory, which isn't usually used.

> Since neither X nor Y are identified, suppose X is the universal set.

> Can there be a disjoint set Y let alone one with cardinality
> k?

What's a disjoint set? That's nonsense.
Can there be a set Y with cardinality k that's disjoint to X?
That's what the problem is - you're going in circles.

> Maybe I saw the answer in other replies but didn't notice it.

Most likely.

> The syntax being used is a bit strange but seems to be often
> used by many.

Your leaps to unwarranted assumptions is more that strange

Date Subject Author
12/3/12 jaakov
12/3/12 forbisgaryg@gmail.com
12/3/12 Aatu Koskensilta
12/3/12 jaakov
12/3/12 Carsten Schultz
12/3/12 jaakov
12/3/12 Aatu Koskensilta
12/3/12 jaakov
12/3/12 Aatu Koskensilta
12/3/12 jaakov
12/3/12 Carsten Schultz
12/3/12 jaakov
12/3/12 Aatu Koskensilta
12/3/12 Butch Malahide
12/3/12 jaakov
12/3/12 Butch Malahide
12/4/12 jaakov
12/4/12 forbisgaryg@gmail.com
12/4/12 William Elliot
12/4/12 forbisgaryg@gmail.com
12/4/12 William Elliot
12/4/12 William Elliot
12/4/12 jaakov
12/4/12 William Elliot
12/4/12 jaakov
12/4/12 Shmuel (Seymour J.) Metz
12/4/12 Spammer
12/4/12 jaakov