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From: David R Tribble on 27 Jul 2010 17:07
If we consider all of the sets having a given object as a member,
is that collection a proper set itself or something more complex,
i.e., a class?

Define Cs(x) as the collection of sets having x as a member:
Cs(x) = { A | x in A }.

For finite set universes, it is obvious that Cs(x) must be a
finite set of sets within that universe. It's not as clear to me
if the same is true of infinite set universes.

-drt
From: David Hartley on 27 Jul 2010 19:36
In message
<ae0e1284-869f-4119-a3af-29fed56195fb(a)a30g2000vba.googlegroups.com>,
David R Tribble <david(a)tribble.com> writes
>If we consider all of the sets having a given object as a member, is
>that collection a proper set itself or something more complex, i.e., a
>class?
>
>Define Cs(x) as the collection of sets having x as a member:
> Cs(x) = { A | x in A }.
>
>For finite set universes, it is obvious that Cs(x) must be a finite set
>of sets within that universe. It's not as clear to me if the same is
>true of infinite set universes.

For any set y, if {x, y} is a set then it is a member of Cs(x). If {x,
y} is always a set, then Cs(x) has a sub-class that's in one-one
correspondence with the universe. So in "standard" set theory (ZFC with
classes) it must be a proper class, i.e. not a set. I don't know about
theories where the universe is a set, finite or otherwise.
--
David Hartley
From: Aatu Koskensilta on 29 Jul 2010 06:39
David R Tribble <david(a)tribble.com> writes:

> If we consider all of the sets having a given object as a member,
> is that collection a proper set itself or something more complex,
> i.e., a class?

It is a proper class.

> Define Cs(x) as the collection of sets having x as a member:
> Cs(x) = { A | x in A }.
>
> For finite set universes, it is obvious that Cs(x) must be a
> finite set of sets within that universe.

What is a "finite set universe"?

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: David R Tribble on 29 Jul 2010 12:49
David R Tribble writes:
>> For finite set universes, it is obvious that Cs(x) must be a
>> finite set of sets within that universe.
>

Aatu Koskensilta wrote:
> What is a "finite set universe"?

I'm no expert, so excuse the inexact wording. I meant a universe
of sets of a finite size, e.g., the universe of sets having no more
than three members. Perhaps this is a specific kind of "model"?
From: David R Tribble on 29 Jul 2010 12:51
David R Tribble writes:
>> If we consider all of the sets having a given object as a member,
>> is that collection a proper set itself or something more complex,
>> i.e., a class?
>

Aatu Koskensilta wrote:
> It is a proper class.

That was my feeling, but is there a specific line of argument
about how we know that? Is there a particular property (or lack
of one) of Cs that renders it a class?
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