# [FOM] Disguised Set Theory "DST"

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Mon Oct 3 19:55:11 EDT 2011

```Dear Bjordal

There is nothing in this theory that can prevent for example
0 from having a successor (as you defined a set that
has 0 and its E-elements as the sole E-elements of it)
of the form x={0,x}, and this in turn can have a successor
y that is an e-element of itself that has only 0 and x as
its E-elements and this can go on infinitely. The problem is that
this will render your alleged infinite set N to be exactly {0}.

Regards

Zuhair

On Mon, 3 Oct 2011 00:23:15 +0200, Frode Bjordal wrote:

>
> Why isn't N well behaved when defined using comprehension
> (i.e. why
> should filters as you invoke in the note you linked to be
> needed for
> infinity)? Define first ?={x:-x=x} and ordinal successor
> intuitively
> with brackets by z'={u:uEzVu=z}. Spell out and write z'Ey
> for exists
> w(wEy&forallq(qEw iff (qEz V q=z))). Define N by
> comprehension and
> get:
>
>    xeN iff (y)(((?Ey & (z)(zEy =>
> z'Ey))=>xEy)
>
> It seems N is well-defined by your comprehension condition.
> Further,
> obviously ?eN. Clearly also not TC(N)e?, so ?EN. Suppose
> aEN. By
> comprehension, (y)(((?Ey & (z)(zEy =>
> z'Ey))=>aEy). By predicate
> logic, (y)(((?Ey & (z)(zEy =>
> z'Ey))=>(aEy=>a'Ey)). This gives
> (y)(((?Ey & (z)(zEy => z'Ey))=>a'Ey). By
> comprehension then a'EN. So
> ?EN and aEN only if a'EN. Thence N is infinite.
>
>

> Frode Bj?rdal
> Professor i filosofi
> IFIKK, Universitetet i Oslo
> www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html

```