[FOM] disguised set theory "DST"

[Frode Bjørdal]

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



2011/10/5 Randall Holmes <m.randall.holmes at gmail.com>:
> Dear FOMers,
>
> In this note I will explain why Frode Bjordal has not succeeded in
> proving Infinity
> or inconsistency in Zuhair al-Johar's proposed set theory DST
> (disguised set theory).
>
> I should say that I am not arguing as an advocate of this very strange
> theory, but as someone
> who has been trying for some time to show either that it is
> inconsistent (as it rather appears
> it ought to be) or to discover what a model might look like.
>
> The simplest exposition of DST is as follows:  it is a first order
> theory with primitive
> relations e (private membership) and = (equality).
>
> Axiom I:  (Az.z e x <-> z e y) -> x=y (extensionality for private membership)
>
> Axiom II:  trans(a) means (Axy.x e y and y e a -> a e a).  x c= y
> means (Az, z e x -> z e y).
> For any x there is a set
> TC(x) (unique by axiom I) such that for all y, y e TC(x) iff y e A
> for every set A such that trans(A) and x c= A.
>
> Definition:  x E y means x e y and not y e TC(x).
> The E relation is called the public membership relation.
>
> Axiom III:  For every formula phi mentioning only E and =, there is a unique set
> {x|phi} such that for all x, x e {x|phi} iff phi.  Note that it is the
> *private* extension
> of {x|phi} that is determined by the formula phi:  this means for
> example that set brackets
> cannot appear except in parameters in an instance of Axiom III.
>
> Always remember that TC(x) is the transitive closure of x with respect
> to the *private* membership relation.
>
> There are variations.  I have suggested including a full class theory
> (so that Axiom II
> would be impredicative class comprehension as in Kelley-Morse; Axiom
> III would have an additional
> restriction in only allowing sets (e-elements) as parameters, and an
> object witnessing
> Axiom III would be a set in the sense of being an e-element.  I also
> note that in a class
> treatment there is no reason that TC(x) might not be a proper class
> for some sets.  I definitely
> think that a useful version of this theory (something whose existence
> I doubt) would need
> some kind of induction principle to support reasoning about transitive
> closures; the full
> class theory would handle this neatly.
>
> I have considered not allowing = in instances of axiom III.  I have
> considered using the supertransitive closure
> (which includes subsets of its elements as well as elements of its
> elements) instead of the transitive
> closure in the definition of E.
>
> For the discussion of Bjordal's posts, the variations are not needed.
>
> It is important to note that while e is nonwellfounded because it has
> cycles (for example {x | x = x} = V
> is the universal set for e and so V e V), the relation E, while also
> non-well-founded
> (a proof of this would take a little work) is acyclic.  In a chain x1
> E x2 E x3 E ... E xn
> we cannot have xn = x1, because we would then have each xi in the
> transitive closure of each other xj,
> as the relation x e TC(y) is transitive, and x[n+1] e TC(xn) is
> excluded for each n by
> the definition of E.
>
> We turn to Bjordal's proposals for an argument for Infinity, which are
> similar to arguments
> that I have attempted.  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.
>
> Bjordal wants to define ordinal successor intuitively by x' = {u | u E
> x or u = x}.  He
> is aware that x' defined thus cannot appear in an instance of Axiom
> III, so he suggests
> a contextual definition of x' E y as "there is a w E y such that the
> public elements of w
> are exactly the public elements of x and x itself".  He then defines N
> as {n | (forall I.0 E I
> and (forall m.m E I -> m' E I) ->  n E I}.
>
> This exact definition fails for a technical reason having to do with
> the choice of quantifier
> in the contextual definition.  There is an even simpler failure:
>  it happens that not all objects have any successors at all (there is
> no V' for example)
> but Im willing to restrict the notion of inductive set to say that
> elements of inductive sets
> which have a successor have some successor in the inductive set, as a
> friendly amendment).
> It happens that 0 has more than one "successor".  The set {0}
> exists and its public extension is the same as its private extension.
> Define W = {x | x E V}
> (this is not the same as V:  its private elements are the public
> elements of V, that is, the
> sets which do not have V in their transitive closure).  {0,W} has the
> same public extension
> as {0}, and also {0,W} E V is true ({0,V} has the same public extension as {0}
> but is not a public member of anything).  Observe that V-{{0}} and
> V-{{0,W}} are both inductive
> sets -- anything which has a successor and is in either of these sets
> has a successor in that set.
> But this means that {0} is not a public element of N.  In fact, for
> any successor 1 of
> 0, V-{1} is inductive in the sense required by Bjordal's definition of
> N, so 1 is not in N.
> It rather appears that Bjordal's N is just {0}.  This set is not
> closed under successor, which
> shows directly that his claims about consequences of his definition are false.
>
> So, I make a further friendly amendment:  change the contextual
> definition of x' E y to
> "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".
>
> 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.
> Now suppose x E N.
> We can deduce that x e N, and so that x belongs to every public
> extension which contains 0
> and is closed under successor in our precise sense, and so any
> successor x' of x also belongs
> to all these public extensions, so for any successor x' of x, we have
> x' e N.  At this point
> in Bjordal's argument, he forgets the distinction between public and
> private membership -- he jumps
> to x' E N.  Unfortunately this does not follow.  We know that N is not
> in the private extension of TC(x)
> because x E N:  so we know that N is not in the transitive closure of
> any of the public elements
> of x'.  But x' might have private elements (private because they have
> x' in their TC) which
> also have N in their TC.  One could protest -- but what about the
> specific x' defined
> by {u | u E x or u = x}?  This has exactly the right private members
> and we can see that
> N is not in its TC.  Unfortunately, it might be the case that the
> public extension of this "x'"
> might not be the same as its private extension:  if x' e TC(x), this
> will happen.  This implies
> that x is a very bad set; unfortunately there does not seem to be any
> way to exclude such bad
> sets from our inductive sets (the conditions that define them involve
> e essentially and cannot
> in any obvious way be expressed in terms of E and =).
>
> It should be noted of course that any model of DST (if there are any)
> is externally infinite; the question
> here is whether DST contains a set which it can tell is infinite in
> internal terms.
>
> In a second post, Bjordal presented an argument for inconsistency of DST:
>
> Here is Bjordal's argument, with my comments in brackets:
>
> Let ordered pairs be defined a la Kuratowski
>
> [<x,y> = {{x},{x,y}} can be defined as a private extension;
> if <x,y> is in the transitive closure of {x,y} it will not have the
> correct public extension; there is
> no way to specify the privately defined <x,y> using just E and =, but
> of course one can write a formula
> expressing the idea that a set p is a Kuratowski ordered pair of x and
> y; such p's will not be unique,
> but can be used to represent relations definable in terms of E and =,
> as long as the objects related
> are capable of being thrice iterated public elements of sets]
>
> Let Uz signify the union set of z
>
> [the public membership relation is nonextensional so this may not
> describe a single object,
> and in terms of the public membership
> there is no guarantee that a set *has* a union, though I do not have a
> counterexample handy.  We can
> write a formula which says "U is a union set of x" in terms of the
> public membership]
>
> Bjordal continues to use the notations for empty set and ordinal
> successor from above.
>
> Let t(a) be the set provided by the comprehension
> {x | (forall y.<0,a> E y and (forall uv.<u,v> E y -> <u',U(v)> E y) -> x E y}}
>
> [because of failures of notation due to nonuniqueness of objects
> mentioned, this is difficult,
> but in fact something like this set can be defined.  The intention is
> to produce a function taking
> each natural number n to the nth iterated union of a.  The problems
> with induction evidenced above
> suggest to us that this is not going to work perfectly, but as will be
> seen I'm willing to grant for
> the sake of argument that this construction does exactly what is wanted.]
>
> Let X be given by the comprehension {x | (forall ny.<n,y> E tc(x) -> not y E x)}
>
> [X is supposed to be the set of all x such that x is not an element of
> the transitive closure of x
> with respect to public membership.  Bjordal does not realize that
> public membership is acyclic in all cases,
> as the subsequent discussion reveals.  It is not permissible to use
> tc(x) in an instance of comprehension
> in this way, because tc(x) is a set with a specified *private*
> extension.  However, I am willing to
> grant that this whole construction works as intended, because the set
> of all x such that the public transitive
> closure of x does not contain x is indeed provided by Axiom III in a
> different way:  it is the universe!!!]
>
> Clearly x e X iff not X e TC(x) [no, x E X iff not X e TC(x), where X = V].
>
> For if xeX and XeTC(x) then xeTC(x),
> and x would be cyclic. [I *think* that Bjordal is here confusing
> public and private membership.
> He thinks that membership in X will exclude x e TC(x).  It doesn't.
> The definition of X, if taken as
> succeeding, excludes members with cycles in public membership,
> which don't exist anyway, as I showed earlier in this note;
> it does not exclude cycles in private membership, whose existence is
> quite hard to describe in terms of E and
> = (quite probably impossible).  If there is a formula in E and = which
> is equivalent to x e TC(x)
> we DO get a paradox in DST; Bjordal is in many ways on the right track.]
>
> As xEX iff xeX and not XeTC(x), we have that
> xEX iff xeX.  [No, as in the actual case X=V we have x e X true in all
> cases but x E X true
> only when X not in TC(x).]
>
> 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.
> [As I have shown above, there are no sets which are cyclic in the
> sense of public membership.]
>
> Suppose next that X is not cyclic.
> [the only possibility]
>
>  Then X fulfills the comprehension
> condition for X and we derive that XEX.
>
> [Just as at the end of his infinity argument,
> Bjordal forgets that comprehension gives *private* extensions.  We can
> only conclude
> X e X -- in the case X=V which actually obtains, we have V e V but we
> do not have V E V
> because V e TC(V) = V.]
>
>  So X is cyclic iff X is not
> cyclic according to the suggested set up.[As we see above, the
> argument fails in various ways.]
>
> Conclusions:  This theory is very strange.  I would like to see its
> status settled:  it seemed to me
> at first blush that it ought to be inconsistent and relatively easy to
> show inconsistent, but it seems not
> to be the case (a great deal of information is concealed by the
> restriction of formulas defining sets
> to those using public membership).  It is also clear that the world of
> DST (if there is such a world) is
> quite weird.  If some FOMer can come up with a model or an
> inconsistency proof I will be very interested,
> but do remember that some very subtle distinctions are at play here.
> The fact that extensions for one membership
> relation are defined in terms of another membership relation is
> reminiscent of the situation in Kisielewicz's
> "double extension set theory", two versions of which I showed to be
> inconsistent in a quite elaborate way,
> though the consistency of the weakest system he proposed remains an
> open problem.  Unlike Zuhair, I do
> not see this theory as a place to actually do mathematics -- it is too
> strange, even if it is consistent.
>
> I'm happy to discuss the other things I know or think I know about
> this theory with anyone who is interested.  I state
> for the record that I have made a fair number of mistakes myself
> trying to think about DST; it is extremely tricky.
>
>
> --
> Sincerely, Randall Holmes
>
> Any opinions expressed above are not the
> official opinions of any person or institution.
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>