[FOM] Disguised Set Theory "DST"

[Frode Bjørdal]

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; 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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Dear Zuhair,

I believe the following may answer your query concerning the
consistency of your suggested disguised set theory in the negative. In
the following I presuppose that the readers have digested the
terminology of the note you linked to in your message.

Let ordered pairs be defined e.g. à la Kuratowski. Let Uz signify the
union set of z. Let z’ signify the ordinal successor of z. Let Ø
signify the empty set {x:-x=x}. Let t(a) be the set provided by the
comprehension {x:(y)((<Ø,a>Ey & (u)(v)(<u,v>Ey=><u’,Uv>Ey))=>xEy)}.
Let X be given by the comprehension {x:(n)(y)(<n,y>Et(x)=>-xEy)}.
Clearly, xeX only if not XeTC(x). For if xeX and XeTC(x) then xeTC(x),
and x would be cyclic. As xEX iff xeX and not XeTC(x), we have that
xEX iff xeX. Suppose first that X is cyclic, i.e. XEX or XE(2)X or…
Then we derive in a finite number of steps that X is not cyclic.
Suppose next that X is not cyclic. Then X fulfills the comprehension
condition for X and we derive that XEX. So X is cyclic iff X is not
cyclic according to the suggested set up.

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