# [FOM] Disguised Set Theory "DST"

Frode Bjørdal frode.bjordal at ifikk.uio.no
Sun Oct 2 18:23:15 EDT 2011

```2011/10/1 Zuhair Abdul Ghafoor Al-Johar <zaljohar at yahoo.com>:
> Dear FOMers.
>
> DST is a theory that I defined in first order logic with equality "=" and
> membership "e". The theory has a scheme that looks paradoxical at first
> glance but proves to be quite evasive. The theory can interpret bounded
> ZF-Power-Infinity;

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 Ø={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)(((ØEy & (z)(zEy => z'Ey))=>xEy)

It seems N is well-defined by your comprehension condition. Further,
obviously ØeN. Clearly also not TC(N)eØ, so ØEN. Suppose aEN. By
comprehension, (y)(((ØEy & (z)(zEy => z'Ey))=>aEy). By predicate
logic, (y)(((ØEy & (z)(zEy => z'Ey))=>(aEy=>a'Ey)). This gives
(y)(((ØEy & (z)(zEy => z'Ey))=>a'Ey). By comprehension then a'EN. So
ØEN and aEN only if a'EN. Thence N is infinite.

can define big sets that ZF can't. However the
> consistency of this theory remains an open question.
>
> The theory is actually very simple. It's an extensional pure set theory,
> every set has a transitive closure "TC" defined as the minimal transitive
> superset, induction for transitive closures is stipulated. A binary
> membership relation E is defined as:
>
> x E y iff (x e y and not y e TC(x))
>
> The comprehension scheme simply states that for every formula phi using
> only predicates = and E, the set {x|phi} exists.
>
> See: http://zaljohar.tripod.com/dst.pdf
>
> Regards
>
> Zuhair
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>

--
Frode Bjørdal
Professor i filosofi
IFIKK, Universitetet i Oslo
www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html
```