[FOM] Disguised Set Theory "DST"

[Zuhair Abdul Ghafoor Al-Johar]

Dear F.Bjordal:

 Your main mistake was mixing up your intended concepts
with your contextual definitions that didn't copy them
fully. Also you mixed up between privet and public
membership relations, which is quite natural really.

By the way, DST was further scrutinized by Holmes
and myself. 

 Regarding making use of sets as provided by
the set brackets in disguised comprehension, this
is allowed when those sets are replacing a *parameter*
in a disguised formula. That's why Holmes's formula
is disguised, since 0 in it is replacing a parameter.
Also your inclusion of 0 in your formula was not
a mistake.

The theory is not bizarre, it is non standard, but
definitely follows a disciplined axiomatic system.
Its axioms pivots around a central theme, unlike
ZF which has an ad hoc looking axiomatic system
although its structure is naturally looking. The
only theory that has a more disciplined looking
axiomatic system is NF actually.

Definitely DST is hard to work with, and a lot of 
confusion between public and privet membership
occurs, and its structure is not that natural,
and for now we don't have a clear idea about
what that structure is, so it has unclear semantics
yes. And no matter what that structure
turns to be it is definitely by far much more
difficult to understand than that of ZF.
All those are drawbacks of DST.

We already know that DST interprets bounded ZF
-Power-Infinity, and if filters were added
(if that was consistent) then it proves bounded ZF.
So it is definitely non trivial.

Regarding weather it would be mathematically
practicable this would largely depend on what
kind of results we can obtain from this theory,
if it is the same stuff we can obtain from
theories having clearer structure, then of course
one would prefer to practice in clearer milieu,
if it come up with new stuff that proves to be
mathematically significant that no parallel
theory with clearer structure provides then it would
be a media for practicing some mathematics.

And even if it turns to be inconsistent, i think
the proof of its inconsistency would be an interesting
work, since DST proves to be very elusive as far
as tripping into a paradox is concerned.

Your correspondence nevertheless was positive,
it opened the door for understanding how to carry
out inductive reasoning in DST, which is not
that straightforward.

Regards

Zuhair



On Wed, 5 Oct 2011 22:43:11 +0200, Frode Bj?rdal wrote:

 
> In two posts above I had a shot at showing that the
> disguised set
> theory (DST) proposed by Zuhair (and as it turned out
> scrutinized by
> Holmel) proves infinity, and next a shot at showing that
> DST is
> inconsistent. The presentation of  DST which I first
> related to was at
> certain ponts confusing to me. Indeed Holmes also admits
> that the
> theory is rather bizarre, as it were. Anyway, it turned out
> that I
> made assumptions in my argument which did not abide by the
> disguised
> comprehension principle presupposed.
> 
> It seems to me that the fundamental mistake in my posts was
> in
> presupposing that we can make use of sets as provided by
> the set
> brackets in disguised comprehension. It is true that I at
> one point
> was making use of a contextual definition, but on this
> matter I was
> wavering.
> 
> I want to point out that Holmes seems to make a similar
> mistake in his
> posting. He there attempts at a rectified definition of my
> N by having
> 0={x:x=I=x}, and defining z'EY contextually as  "for
> every w such that
> the public elements of w are exactly the public
> > elements of x
> > and x itself, and such that w is a public element of
> some set, w E y". More formally: z'Ey iff
> (w)((s)(sEw<=>sEzVs=z)=>wEy).
> 
> His N is now supposed to be defined by the condition
> (y)(0Ey&(z)(zEy=>z'EY)=>xEy).
> 
> Concerning this Holmes in his post remarks:
> 
> > This set is in my opinion defined correctly. ?But it
> does not have
> > nice properties. ?Clearly
> > 0 e N, and since N is obviously not in TC(0), we have
> 0 E N as well.
> 
> However, the condition
> (y)(0Ey&(z)(zEy=>z'EY)=>xEy) is not a disguised
> condition as I have now come to understood this.
> 0={x:x=I=x} occurs.
> On this very set Holmes in the same post earlier remarks:
> 
> Define 0 as {x | x =/= x}, the empty set.
> > Note that this is *the*
> > empty set only in relation to private membership. ?The
> set {V} for
> > example whose only private
> > element is the universe, is readily seen to also have
> no public elements.
> >
> 
> If one should attempt at defining N along such lines as
> Holmes are on
> to here it should, I believe, be by using the existence of
> publically
> empty sets as initial, i.e. something like this:
> 
> 
> N={x:(y)((existsw(wEy&-existst(tEw))&(z)(zEy=>z'Ey))=>xEy)}
> 
> I have no intuition concerning the nature of N in DST. It
> seems to me
> that DST is sufficiently foreign to intuition that I am
> strongly
> inclined to agree with Holmes in that it is too strange to
> be
> mathematically practicable even if consistent. However, it
> would be
> interesting if something non-trivial was found out.
> 
> -- 
> Frode Bj?rdal
> Professor i filosofi
> IFIKK, Universitetet i Oslo
> www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html