Thread View: sci.logic
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Started by Bill Taylor
Wed, 04 Nov 1992 03:16
Re: Impredicativity - was: Russell's Paradox
Author: Bill Taylor
Date: Wed, 04 Nov 1992 03:16
Date: Wed, 04 Nov 1992 03:16
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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 ? :-(
------------------------------------------------------------------------------
Bill Taylor wft@math.canterbury.ac.nz
------------------------------------------------------------------------------
Well, I believe in solipsism, but that's just one man's opinion.
------------------------------------------------------------------------------
Re: Impredicativity - was: Russell's Paradox
Author: Vaughan R. Pratt
Date: Wed, 04 Nov 1992 07:37
Date: Wed, 04 Nov 1992 07:37
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In article <Bx693z.H37@cantua.canterbury.ac.nz> wft@math.canterbury.ac.nz (Bill Taylor) writes:
>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
What is the reasoning here? That a theory T "intrinsically suggests"
not-P for any P not in T? If neither P nor not-P are in T, which of
not-P or not-not-P (= P) does T "intrinsically suggest?"
>The slick form of AF, that every nonempty set has a member disjoint from it,
Where is this form given? And why doesn't it rule out Omega = {Omega}?
(Just asking, I'm not expert on this stuff.)
>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 !!!!
If you are comfortable with A, you should not mind
B = { 1 , { 1 , { 1 , { 1 , {.....}}}}} .
But by AFA, B = {1, B}, making B a member of itself.
As soon as you weaken FA to AFA, the possibility of self-membership
arises.
>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 ? :-(
I do. But I forgot to say in my earlier message that the main point of
my positive reformulation of Russell's paradox was to remove this
impredicativity from the argument. Witness Y was constructed as the
set of all *elements of X* not members of themselves. This established
a positive result without any impredicativity. Russell's theorem is
not so much a corollary of this theorem as an interpretation of it, in
that "there is no set of all sets" formalizes as "every set is
nonuniversal", or "every set has a nonmember".
>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.
ZF+FA is even stronger than ZF+AFA. So by that reasoning ZF+FA can't
look like like a fundamental set theory either. Yet you seem to prefer
ZF+FA to ZF+AFA.
>I hope you are taking your own rubbishing
>(concerning Z vs ZF) like a man ! It's a tough life on the net !
Being mistaken at the top of your voice does have its drawbacks.
Yesterday definitely wasn't my day, I got something really wrong on
sci.math involving measure theory, much more wrong than Z vs. ZF, which
I maintain is a highly debatable matter, the near-universal acceptance
of the cumulative hierarchy as a model of ZF notwithstanding.
--
Vaughan Pratt There's no truth in logic, son.
Re: Impredicativity - was: Russell's Paradox
Author: Michael Zeleny
Date: Wed, 04 Nov 1992 18:45
Date: Wed, 04 Nov 1992 18:45
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In article <1992Nov4.073717.22625@CSD-NewsHost.Stanford.EDU>
pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <Bx693z.H37@cantua.canterbury.ac.nz>
>wft@math.canterbury.ac.nz (Bill Taylor) writes:
>>[...]
>[...]
BT:
>>The slick form of AF, that every nonempty set has a member disjoint from it,
VP:
>Where is this form given?
Most everywhere, e.g. in Drake (_Set Theory_), whose pithy comments on
the Axiom of Replacement (a.k.a. the `F' in `ZFC') will go a long way
toward disabusing you of your wacky revisionist notions. The standard
discussion is to be found in Fraenkel, Bar-Hillel, and Levy,
_Foundations of Set Theory_.
VP:
> And why doesn't it rule out Omega = {Omega}?
>(Just asking, I'm not expert on this stuff.)
Neither am I, but that will not prevent me from being charitable
enough to let you answer your own question.
>>[...]
>[...]
BT:
>>I hope you are taking your own rubbishing
>>(concerning Z vs ZF) like a man ! It's a tough life on the net !
VP:
>Being mistaken at the top of your voice does have its drawbacks.
You have my hard-earned sympathies.
VP:
>Yesterday definitely wasn't my day, I got something really wrong on
>sci.math involving measure theory, much more wrong than Z vs. ZF, which
>I maintain is a highly debatable matter, the near-universal acceptance
>of the cumulative hierarchy as a model of ZF notwithstanding.
Some _philosophical_ reservations concerning Replacement can be made
from a generic Limitation of Size standpoint, -- right now I do not
have the references, but they are likely to be found in Maddy's JSL
expository articles on the ZFC axioms. However, from the more refined
standpoint of the cumulative hierarchy, Drake more or less has the
last word.
>--
>Vaughan Pratt There's no truth in logic, son.
I'll take it under advisement, pops.
N.B. The Phallogocentric Cabal meets each Tuesday at Emerson 307.
Next week: the redoubtable Bill White explicates the hidden agenda
behind Boolean-valued models of set theory.
cordially,
mikhail zeleny@husc.harvard.edu
" -- I shall speak bluntly, because life is short."
P.S. Will trade my spare copy of Bell's topos theory book for Jech or
WHY. Also have a spare copy of Lambek and Scott. Always looking for
logico-philosophical esoterica, Italian motorcycle parts, Swiss
firearms, or XVIIIth century French smut.
Re: Impredicativity - was: Russell's Paradox
Author: Vaughan R. Pratt
Date: Thu, 05 Nov 1992 00:47
Date: Thu, 05 Nov 1992 00:47
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In article <1992Nov4.134534.17092@husc3.harvard.edu> zeleny@husc10.harvard.edu (Michael Zeleny) writes:
>
>Taylor:
>>>The slick form of AF, that every nonempty set has a member disjoint from it,
>
>Pratt:
>>Where is this form given?
>
>Most everywhere [...]
>
>>Pratt: And why doesn't it rule out Omega = {Omega}?
>>(Just asking, I'm not expert on this stuff.)
>
>Neither am I, but that will not prevent me from being charitable
>enough to let you answer your own question.
Why, do you turn in all your Usenet posts along with your tax return?
Actually it's just me being dense (and Bill being nonstandard). I took
Bill's AF to mean AntiFoundation when he meant Axiom of Foundation
(whence he *meant* to rule out Omega={Omega}). I should have figured
that out sooner (broken enough eggs in the past two days to start an
omelette parlor).
>[...] e.g. in Drake (_Set Theory_), whose pithy comments on
>the Axiom of Replacement (a.k.a. the `F' in `ZFC') will go a long way
>toward disabusing you of your wacky revisionist notions.
Thank god. I've been posting wacky messages for weeks and no one seems
to have noticed till now. How do you do it?
But let's get our facts straight on what I'm proposing to revise. I
haven't lifted a finger against Replacement, in fact I need it to make
money with it by 2012 (come on, Replacement, come on, be strong for
me!).
>Some _philosophical_ reservations concerning Replacement can be made
>from a generic Limitation of Size standpoint, -- right now I do not
>have the references, but they are likely to be found in Maddy's JSL
>expository articles on the ZFC axioms. However, from the more refined
>standpoint of the cumulative hierarchy, Drake more or less has the
>last word.
Again, I'm not in the market to be talked out of Replacement, or talk
anyone out of it. Foundation on the other hand seems as pointless as
the cumulative hierarchy: each needs the other and neither seems to be
much use to the rest of mathematics. Now who can you point me at with
a convincing pitch for Foundation?
>Always looking for
>logico-philosophical esoterica, Italian motorcycle parts, Swiss
>firearms, or XVIIIth century French smut.
Oh. Guess that answers my second last question. :-)
--
Vaughan Pratt There's no truth in logic, son.
Re: Impredicativity - was: Russell's Paradox
Author: Torkel Franzen
Date: Thu, 05 Nov 1992 09:08
Date: Thu, 05 Nov 1992 09:08
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In article <1992Nov5.004725.8252@CSD-NewsHost.Stanford.EDU> pratt@Sunburn. Stanford.EDU (Vaughan R. Pratt) writes: >Again, I'm not in the market to be talked out of Replacement, or talk >anyone out of it. Foundation on the other hand seems as pointless as >the cumulative hierarchy: each needs the other and neither seems to be >much use to the rest of mathematics. The point of the cumulative hierarchy is that it gives an interpretation of ZF, as Zermelo lucidly explains in his 1930 article. The whole business of "Down with foundation!", "Oppose unnatural self-membership!", "Give me replacement or give me death!" as it has appeared here is completely bogus. If you want to introduce a new set theory or a new interpretation of existing set theory, by all means do so if you can. What I am interested in is whether or not you can explain what you are talking about, as has been done with the "standard" theory of non-well-founded sets (in that case by giving a model of the theory in ordinary mathematics).
Re: Impredicativity - was: Russell's Paradox
Author: Randall Holmes
Date: Thu, 05 Nov 1992 16:42
Date: Thu, 05 Nov 1992 16:42
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In article <1992Nov4.073717.22625@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <Bx693z.H37@cantua.canterbury.ac.nz> wft@math.canterbury.ac.nz (Bill Taylor) writes:
>>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
>
>What is the reasoning here? That a theory T "intrinsically suggests"
>not-P for any P not in T? If neither P nor not-P are in T, which of
>not-P or not-not-P (= P) does T "intrinsically suggest?"
>
>>The slick form of AF, that every nonempty set has a member disjoint from it,
>
>Where is this form given? And why doesn't it rule out Omega = {Omega}?
>(Just asking, I'm not expert on this stuff.)
Thomas Jech, _Set Theory_, first page. It does rule out Omega =
{Omega}; it is a form of the axiom of foundation (AF, not AFA).
>
>>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 !!!!
>
>If you are comfortable with A, you should not mind
>B = { 1 , { 1 , { 1 , { 1 , {.....}}}}} .
>
>But by AFA, B = {1, B}, making B a member of itself.
This uses a very strong extensionality property which follows from
AFA; there is no reason why there cannot be a sequence of distinct
sets xi such that xi = {1,x[i+1]} in ZFC without Foundation. Of
course, this makes the symbol above ambiguous (the original expression
with 1,2,3,4... also fails to specify a unique set in ZFC- -- there
could be many such sets).
>
>As soon as you weaken FA to AFA, the possibility of self-membership
>arises.
You do not "weaken" FA to AFA; ZFC- + FA and ZFC- + AFA are both
stronger than ZFC-, but neither is stronger than the other.
>
>>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 ? :-(
>
>I do. But I forgot to say in my earlier message that the main point of
>my positive reformulation of Russell's paradox was to remove this
>impredicativity from the argument. Witness Y was constructed as the
>set of all *elements of X* not members of themselves. This established
>a positive result without any impredicativity. Russell's theorem is
>not so much a corollary of this theorem as an interpretation of it, in
>that "there is no set of all sets" formalizes as "every set is
>nonuniversal", or "every set has a nonmember".
>
>>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.
It doesn't; its consistency can be proven in ZFC- (or ZF-) (just like
that of full ZFC). Foundation or the lack thereof has nothing to do
with consistency strength.
>
>ZF+FA is even stronger than ZF+AFA. So by that reasoning ZF+FA can't
>look like like a fundamental set theory either. Yet you seem to prefer
>ZF+FA to ZF+AFA.
No, it isn't. They have exactly the same strength (and both describe
hierarchically structured universes).
>
>>I hope you are taking your own rubbishing
>>(concerning Z vs ZF) like a man ! It's a tough life on the net !
>
>Being mistaken at the top of your voice does have its drawbacks.
>Yesterday definitely wasn't my day, I got something really wrong on
>sci.math involving measure theory, much more wrong than Z vs. ZF, which
>I maintain is a highly debatable matter, the near-universal acceptance
>of the cumulative hierarchy as a model of ZF notwithstanding.
>--
>Vaughan Pratt There's no truth in logic, son.
--
The opinions expressed | --Sincerely,
above are not the "official" | M. Randall Holmes
opinions of any person | Math. Dept., Boise State Univ.
or institution. | holmes@opal.idbsu.edu
Re: Impredicativity - was: Russell's Paradox
Author: Vaughan R. Pratt
Date: Thu, 05 Nov 1992 17:45
Date: Thu, 05 Nov 1992 17:45
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In article <1992Nov5.164251.29649@guinness.idbsu.edu> holmes@garnet.idbsu.edu (Randall Holmes) writes: >You do not "weaken" FA to AFA; ZFC- + FA and ZFC- + AFA are both >stronger than ZFC-, but neither is stronger than the other. I guess this must be another of my misconceptions. My general impresssion was that AFA was automatically true under FA, by virtue of the latter saying that only "unwound" graphs were permitted, and that every chain on the unwinding had to terminate. So what's an example FA situation contradicting AFA? -- Vaughan Pratt There's no truth in logic, son.
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