🚀 go-pugleaf

Many thanks to those who responded to my impredicativity article.

Vaughan R. Pratt says...

>Ooh, ouch, you got toe number 3.  :-)

Sorreeeee !

He also suggests my views may be due to "an accident of culture", by which
he probably means a form of Zermelist indoctrination.  ;-)
However, if I may be permiited to know something of my own mind, I strongly
doubt it. It's true no one can escape their enculturation. However, I can
still distinctly recall my feelings (back in another age of the world!), upon
first hearing Russell's paradox, Russell's barber, the liar paradox, the
heterological paradox and maybe others. This was about high school/undergrad
transition time, and I'd certainly never heard of anything like a hierarchy,
stratification, or anything else. Barely even of set theory, (yes, it really
*was* a long age ago!)  So I can safely say my "resolution" had nothing to
do with Zermelist thought.

As I say, my immediate thoughts were (i) that these paradoxes were basically
all the same, (though now some are pompously distinguished as being semantic
or syntactic); and (ii) that the resolution was obvious, namely that the
paradoxes were a simple cheat caused by defining something in terms of itself,
a vicious circle, in fact. (So I was very pleased later to see Russell himself
had used this term.)

Since then, like us all, I have read masses on this and related subjects, but
my views on it  have changed little. The main change has been my (doubtless
limited) increase in understanding of the idea of impredicativity in general.
I have been saddened that the Russell/Poincare ideas were so readily
dismissed by confusing the vicious circle with impredicativity in general.
That's why I went to the trouble of carefully formulating my view, which I
gave in my article, as to why harmless impredicativity and essential
impredicativity were so different; and *clearly* different, I hoped to show.

No-one has really responded to my idea though; that the "impredicativity of
self-reference" and the "impredicativity of self-candidacy" were clearly
distinguishable.   Pity.  *I* thought it was a good idea. (sob)

I have given this long boring psycho-history partly to refute the charge
that I was merely enculturated (no flame intended, Vaughan, I liked your
article); and partly to back up a further denial. Namely this view is NOT
just acceptance of well-foundedness, as Vaughan further suggests. In fact,
though I think that "real" sets may well be well-founded, I don't necessarily
insist on it. One should exclude self-membership (as being vicious); but
this can be done in various ways. Indeed, ZF intrinsically suggests it
*without* the bother of a foundation axiom, by limiting the set formation
possibilities so that self-membership is at least not immediately achieved.
The slick form of AF, that every nonempty set has a member disjoint from it, 
may well be true, but it's too close (in its "non-definitional" form) to AC
for my taste, and I suspect AC is definitely false. The more revealing form
of AF, that every set has a rank, is even more suspicious, as it looks even
closer to AC (in the form every set has (ordinal) cardinality). So no; I'm
not madly in favour of AF. The good old ZF axioms look basic to me, and no
more; these have the proper format of only specifying a set to exist if a
formula for its members can be given; that's all a "definitionalist" like
me can truly believe in.

But I *still* think self-membership is the basic paradoxical idea, at least
if membership is to be interpreted in the naive way.

But I'm open on well-foundedness: for all I know, there may well be sets like
A = { 1 , { 2 , { 3 , { 4 , {.....}}}}} . Such a set seems very well defined on
the face of it, so it *could* exist, for all I know. No self-membership there.

REQUEST: If someone can produce a way of getting a self-membered set from the
set A above, using the other ZF constructions, I would *love* to see it !!!!

The real advantage of the iterative hierarchy, and its well-foundedness, is
that it seems to make clear that ZF *really is* consistent. As sure as it
can be short of a proof. It would be much harder to have confidence in the
consistency of a system whose only models had cycles of membership, or
other "mixed-up" features. That's partly why I don't think that Aczel's 
set theory is really basic, delightful though it is. It can't look like
fundamental set theory if a convincing show of consistency has to route
through ZF's consistency. This is just personal prejudice though.

But anyway, Vaughan, thanks for your response. I especially liked your
"positive" version of the Cantor-Russell theorem. It's *always* nice to
have something expressed in positive form; I wish the math world would
make moves to get rid of words like "non-negative", "non-increasing", and
so on. I've always been irked by these. In my book, they are non-words.   ;-)

>For my own part I cannot imagine how I ever used to believe that FA was
>an inherent property of sets.  I was simply bullied into it.

Poor old thing !  You stick to your guns !!

>>                    Kleeneness is next to Godelness.
> Dynamite .sig!
 
Thanks ! I liked yours, too.  Non-reversing letters indeed !  GROAN.  :)

I didn't realize you were an Oz, Vaughan, it doesn't show up in your sig.
But it explains why I've always found your postings quite familiar in tone.
I only discovered you were Oz from your letter giving a (well-deserved)
rubbishing to Robert Silverman.  I hope you are taking your own rubbishing
(concerning Z vs ZF) like a man !  It's a tough life on the net !


Many thanks too, to David Gudeman, for his long and interesting article,
clarifying his views somewhat on impredicativity vs recursiveness, and
sets vs properties. David ascribes great importance to the distinction
between properties and sets, as far as sorting out the Russell paradox goes.
Indeed, they may well be different things; many have thought so, but most
have had a hard time making much headway with the idea. Many mathematicians
probably believe that there are many sets which correspond to no property,
("random-ish" ones); and perhaps that there are different properties
corresponding to the same set (intensional definitions of sets). But neither
idea seems to have become settled into orthodox math.

And I don't really think Russell is altered by framing it in property form:
  "the property of being a property which doesn't apply to itself".
This seems to me essentially the same as the other forms; and similarly
resolvable by my naive intuition described above. Namely, that one cannot
make any sense of the idea that any property could even *possibly* apply
to itself, or it won't have been defined at all; vicious circle again.

The trouble with David Gudeman's idea, that some "apparent" properties are
not really properties, is that he gives no clear way of determining when
they are and when they're not.


That is another reason I went to the trouble of posting my first article; it
seems to me to make a very clear criterion for locating the essential
impredicativity of Russell, which shows up most clearly in the *set* form:
                 { x | P(x) }  ,
the essential impredicativity comes from the bad range of x, not the form of P.

Doesn't *anyone* else think so ?   :-(

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            Bill Taylor              wft@math.canterbury.ac.nz
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       Well, I believe in solipsism, but that's just one man's opinion.
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