[FOM] Disguised Set Theory "DST"

Zuhair Abdul Ghafoor Al-Johar zaljohar at yahoo.com
Tue Oct 4 04:58:58 EDT 2011

Dear F.Bjordal.

I think that t(a) represent a set with E-elements <0,a>,<1,Ua>,<2,UUa>...
which is not provable in this theory yet, but even if we suppose
it exists then your set X is the set of all sets x such that
not x E x and not x E2 x and not x E3 x ..........
But by then X is simply the set of all sets, i.e. X=V,
since all sets in this theory has this property.
You said the following:
"Clearly xeX only if not XeTC(x)" which is false, and
your argument that x would be cyclic doesn't matter
since x can be cyclic (if what you mean is e-cyclic)
and yet it is not an Ei member of itself, V is an obvious
example. Mind you that all sets are hereditarily E-acyclic.
In the last line of your argument you said we derive
that X E X, which is false we derive X e X, which
is not a problem since X is V after all an indeed
it is E-acyclic and indeed it has itself as an e-element
of itself, no problem at all. In the line before it
you said suppose that X E X or X E2 X i.e. you meant X is E-cyclic
but in this theory there is no such X.



On Mon, 3 Oct 2011 05:20:30 +0200, Frode Bjordal wrote:

> 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. ? la Kuratowski. Let Uz
> signify the
> union set of z. Let z? signify the ordinal successor of z.
> Let ?
> signify the empty set {x:-x=x}. Let t(a) be the set
> provided by the
> comprehension {x:(y)((<?,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 Bj?rdal
> Professor i filosofi
> IFIKK, Universitetet i Oslo
> www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html


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