Date: Thu, 1 Apr 1999 13:10:47 +0100 (BST)
From: Paul Taylor
Subject: categories: Is Zermelo-Fraenkel set theory inconsistent?
IS ZERMELO-FRAENKEL SET THEORY INCONSISTENT?
At the end of this message is a sketch of an argument that leads to
the conclusion that Zermelo-Fraenkel set theory is inconsistent.
The impact on mathematics is not as devastating as the incautious
observer might suppose. Recall that ZERMELO set theory (1908), which
is essentially equivalent to the categorists' notion of ELEMENTARY TOPOS
with natural numbers and the axiom of choice, is adequate for
most of the purposes of mathematics, though not, as I shall try to
explain, logic (and theoretical computer science).
ZERMELO-FRAENKEL set theory is the extension of this system by the
axiom-scheme of REPLACEMENT, which was first formulated by Adolf
(later Abraham) Fraenkel, Nels Lennes and Thoralf Skolem in 1922,
although Dimitry Mirimanoff already had something of the idea in 1917.
Notice that this is some two decades after the appearance of the famous
"antinomies" of set theory, so presumably the set theorists' guard had
dropped by that time, and they had begun again to assert megalomaniac
axioms. On the other hand, it is a decade before the second generation
of paradoxical results, Godel's incompleteness theorem and Turing's
unsolvability of the Halting Problem.
Whenever I see set theory books in a library or bookshop, I turn to
the index to find out what they have to say about Replacement. Usually
there is some trivial result, such as the existence of what categorists
call image factorisation, that could have been proved from Zermelo's
axioms with a little more facility in set-theoretic constructions.
The basic use of Replacement, that you will find in the better set theory
books, is the recursive construction of sets (in substance -- types or
objects to type-theorists and categorists -- rather than their names).
For example, Mostowski's theorem states that every well founded extensional
binary relation < is isomorphic to the membership relation for a unique
transitive set. This is found, recursively, by means of the formula
f(x) = { f(y) | y < x },
which also provides the extensional reflection (quotient) of any well
founded relation. In fact the latter result (where the quotient relation
is merely another relation, rather than a membership relation) can be
proved using the topos or Zermelo axioms alone, and not Replacement [1],
although there are categorical generalisations of this that certainly do
need Replacement.
Richard Montague [2] proved a result that should have been taken as a
warning of the perilous nature of Replacement, though I suspect that
Montague's personal eccentricities may have been the reason why he was
ignored. ZF can prove the consistency, not only of Zermelo set theory (Z)
itself, but also of Z extended by any single theorem of ZF.
Adrian Mathias has claimed [3] that Bourbaki was "ignorant" of Replacement,
ie that it did not occur in "Theorie des Ensembles" [4]. Although
Bourbaki is hardly very clear on this matter, it does include a
version of Replacement in its axioms, indeed one that is in widespread
use in category theory and other parts of mathematics, namely that one
can form the UNION of any SET-INDEXED FAMILY of SETS.
One application of N-indexed unions in theoretical computer science is
Scott's "D-infinity" construction of models of the untyped lambda calculus.
Starting from any domain D0=D, one may form its function-space D1=(D0->D0),
and similarly D2=(D1->D1), etc., linking these together with embedding-
projection pairs. If D was one of the examples of L-domains, having a
pair of elements with infinitely many minimal upper bounds, then one can
show (classically) that D-infinity has the cardinality of a model of
Zermelo set theory, so need not exist within such a model unless it also
satisfies Replacement.
These two ways of seeing Replacement have a common theme: we use N-indexed
or transfinite unions to unfold a free(ish) model of one logic within
a model of another.
Having seen this in the context of a messy domain-theoretic construction,
we may think in a more disciplined way about free models of the lambda
calculus, the topos axioms, etc. In fact, there is no difficulty in
constructing these models, as they are merely TERM ALGEBRAS. The problem
lies in proving that the term algebra has the universal (initiality)
property that qualifies it as "free":
Let S be the universe (the category of ZF-sets, for example)
and F the term algebra (internal to S) for the logic L.
Suppose that S itself is a model of L.
Then there is a unique interpretation functor []:F->S
that takes each syntactic operation of F (eg prod(a,b))
to the semantics ([a] x [b]) in S.
It is merely unique up to unique isomorphism if the L-structure in S is
defined by universal properties rather than being chosen.
This initiality property may also be expressed type-theoretically.
Per Martin-Lof [5] introduced objects with such a property, called UNIVERSES,
observing the analogy with Replacement. This point of view stresses that
the above property is a RECURSION SCHEME.
Let me explain how I came to realise that the existence of []:F->S depends,
in general, on Replacement.
There is an amazingly simple but incredibly powerful argument, due to
Peter Freyd and known variously as (Artin-Wraith) glu(e)ing, sconing,
the Freyd cover, logical relations and other names. It is based on some
very elementary categorical investigations of a certain comma category
involving F and S. This argument has been developed rather a long way
(the most recent paper that I know of is [6]), and we are pretty close
to having a purely categorical proof of the strong normalisation theorem
for lambda calculi that, unlike the syntactic proofs, is completely
generic with regard to the calculus in question.
Freyd originally showed that the terminal object (1) of the free topos (F)
is projective, and more generally the "global sections functor"
F(1,-) : F -> S preserves colimits. In particular, it preserves the
initial object (0), which is categorical jargon for saying that S proves
the consistency of F, because the S-set of F-morphisms 1->0 is the
initial (empty) S-set.
I found this suspicious, because the punch-line of Andre Joyal's 1973
(but as yet unpublished and unavailable) categorical proof of Godel's
incompleteness theorem is that such a functor F(1,-) : F -> S does *not*
preserve the initial object.
The more careful amongst categorists ought also to be suspicious when
I speak of "a functor F(1,-) or [] : F -> S" where F is an INTERNAL
category in S. The meaning that we must give to this phrase is that
it is "syntactic sugar" for a certain FIBRATION p: V -> F, where V is
also an internal category and p and internal functor in S.
This brings us back to the relationship between Replacement as a recursive
construction of objects and Replacement as infinitary colimits:
"p: V -> F" is the colimit (in a 2-category whose objects are fibrations)
of a recursively defined diagram vaguely similar to that which gives Scott's
D-infinity.
I have come to the conclusion that attempts to define "colimits" such as
this are inherently circular: what, after all, does it mean to have a
"cocone" to test such an alleged colimit?
My categorical formulation of Replacement speaks about fibrations
and smaller colimits defined internally in the style of Benabou.
This is to be found in the final section (9.5) of my book [7],
Section 7.7 of which also gives an account of Freyd's gluing construction.
This book is officially due to be published in mid-May, but it is
already in stock (and I have my own copy in front of me), and is
available direct from the publishers at 50 pounds (inclusive of overland
postage and packing). Please contact
Richard Knott,
email: rknott@cup.cam.ac.uk
fax: +44 1223 315 052
tel: +44 1223 325 916 (but other methods are preferable)
snail: Cambridge University Press, The Edinburgh Building,
Shaftesbury Road, Cambridge, CB2 2RU, UK
with your address and credit card number. (2.50 pounds extra for airmail.)
Having seen that Replacement provides a UNIFORM way of proving
consistency of any fragment of logic, we come at last to the
inconsistency argument:
Let L(0) be Zermelo set theory
(or the axioms for an elementary topos).
For each n, let L(n+1) be L(n) plus
as much of the axiom-scheme of replacement as is needed
to justify the gluing construction that shows that
L(n+1) |- ``L(n) is consistent.''
Now let L(infinity) be the union of L(n) over n:N.
If L(infinity) |- false then L(n) |- false for some n.
But L(infinity) |- ``L(n) is consistent,''
so L(infinity) proves its OWN consistency,
contradicting Godel's theorem.
However, L(infinity) has a standard non-trivial interpretation
in Zermelo--Fraenkel set theory, which is therefore inconsistent.
[1] Paul Taylor, Intuitionistic Sets and Ordinals, JSL 61 (1996) 705-44
[2] Richard Montague, Fraenkel's Addition to the Axioms of Zermelo,
pp 91--114 of Bar-Hillel, et al., eds., Essays on the Foundations of
Mathematics, Magnes Press, Hebrew University, 1966 (distributed
by Oxford University Press).
[3] Adrian Mathias, The Ignorance of Bourbaki, Mathematical Intelligencer,
14 (1992) 4-13.
[4] Nicolas Bourbaki, Elements de Mathematique XXII: Theories des Ensembles,
Livre I, Structures, Hermann, 1957 (English translation 1968).
[5] Per Martin-Lof, An Intuitionistic Theory of Types: Predicative part,
pp 73--118 in Rose and Sheperdson, eds., Logic Colloquium '73,
North-Holland, Studies in Logic and the Foundations of Mathematics #80,
1975
[6] Djordje Cubric, Peter Dybjer and Philip Scott, Normalisation and the
Yoneda Embedding, MSCS 8 (1998) 153--192.
[7] Paul Taylor, Practical Foundations of Mathematics, Cambridge University
Press, Cambridge Studies in Advanced Mathematics #59, xii+572pp, 1999.
http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/
Paul Taylor 19990401
This message may be copied elsewhere,
ON CONDITION that it is quoted in its ENTIRETY.
Date: Thu, 01 Apr 1999 12:23:09 -0800
From: Dusko Pavlovic
Subject: categories: Re: Is Zermelo-Fraenkel set theory inconsistent?
> Let L(0) be Zermelo set theory (or the axioms for an elementary topos).
>
> For each n, let L(n+1) be L(n) plus
> as much of the axiom-scheme of replacement as is needed
> to justify the gluing construction that shows that
>
> L(n+1) |- ``L(n) is consistent.''
>
> Now let L(infinity) be the union of L(n) over n:N.
>
> If L(infinity) |- false then L(n) |- false for some n.
>
> But L(infinity) |- ``L(n) is consistent,''
>
> so L(infinity) proves its OWN consistency,
> contradicting Godel's theorem.
>
> However, L(infinity) has a standard non-trivial interpretation
> in Zermelo--Fraenkel set theory, which is therefore inconsistent.
i think there is a gap is in the step
L(infinity) |- "L(n) is consistent"
so L(infinity) proves its OWN consistency
formalized in a suitable category of theories and interpretations, paul's
construction, if i understand it correctly, refers to the colimit of the tower
L(0) --> FL(0) --> FFL(0) -->...
where FX = X + replacement_X, so that FX |- "X is consistent".
IF the colimit of this tower, paul's L(infinity), were a fixpoint of F, THEN it
would indeed prove its own consistency. but it doesn't seem to be a fixpoint:
the restrictions of the replacement to L(0), FL(0) etc. do not imply the
replacement for L(infinity).
btw, i bought paul's book and warmly recommend it.
-- dusko
Date: Thu, 01 Apr 1999 07:52:01 -0800
From: Mike Oliver
Subject: categories: Re: Is Zermelo-Fraenkel set theory inconsistent?
Paul Taylor wrote:
> Now let L(infinity) be the union of L(n) over n:N.
>
> If L(infinity) |- false then L(n) |- false for some n.
>
> But L(infinity) |- ``L(n) is consistent,''
>
> so L(infinity) proves its OWN consistency,
> contradicting Godel's theorem.
How do you conclude, from the fact that L(infinity) |- "L(n) is consistent", that
L(n) is in fact consistent?
Generally, if T1 |- "T2 is consistent", then to conclude "T2 is consistent",
we use the following argument: Suppose T2 is inconsistent. Then
there is some proof by which T2 |- false. Assuming T1 is strong enough
to formalize the deductive system being used, then it follows
that T1 |- "T2 is inconsistent". But by hypothesis, T1 |- "T2 is consistent",
therefore T1 is inconsistent.
But this is not a contradiction unless we were already *assuming* the
consistency of T1 ! I.e. it follows from T1+Con(T1) that
if T1 |- Con(T2), then Con(T2), but it does *not* in general
follow from T1 alone.
So the step from
L(infinity) |- "L(n) is consistent"
to
L(n) is consistent
cannot be formalized in any obvious way in L(infinity), and therefore
you cannot (again in any obvious way) conclude
L(infinity) |- "L(infinity) is consistent."
--
Disclaimer: I could be wrong -- but I'm not. (Eagles, "Victim of Love")
Finger for PGP public key, or visit http://www.math.ucla.edu/~oliver.
1500 bits, fingerprint AE AE 4F F8 EA EA A6 FB E9 36 5F 9E EA D0 F8 B9
From: Thomas Streicher
Subject: categories: Re: incompleteness of ZF
Date: Sat, 3 Apr 1999 17:12:29 +0200 (MESZ)
Dear Paul,
a short reply to your mail about inconsistency of replacement.
As Paul has already remarked in his mail the type-theoretic counterpart of
replacement is that of universe `a la Martin-Loef. As in case of replacement
the use of universes is that they allow for construction of families of types
(as e.g. needed for inverse limit constructions in Domain Theory which cannot
be performed in pure topos logic precisely for this reason).
But as already observed in a survey article by Thierry Coquand
(ftp://ftp.cs.chalmers.se/users/cs/coquand/meta.ps.Z) and, surely,
known to Martin-Loef himself it holds that in type theory with n+1 universes
one may prove the consistency of type theory with n universes simply by
constructing a model using the n+1st universe. But, of course, this extra
universe is needed for the consistency proof. Accordingly, one cannot prove
the consistency of type theory with $\omega$ universes without postulating an
$\omega$th universe.
Quite the same phenomenon is going on in set theory as already pointed out by
some previous replies. However, in set theory due to the presence of
``impredicative'' axioms the proof theoretic strength is incredibly stronger
than set of Martin-Laoef type theory with $\omega4 universes.
Best, Thomas
Date: Tue, 6 Apr 1999 17:41:04 -0400 (EDT)
From: F W Lawvere
Subject: Re: categories: Re: incompleteness of ZF
Using an old logician's trick (see eg Feferman on paths thru O, or even
Goedel's original papers) as an
April Fool joke
may be amusing to some within the closed gates of a British University,
but is irresponsible on the world network. Think of the hundreds of
lurkers (who hesitate to speak up so that misconceptions can
be discussed and clarified openly, but) who are now furthering the rumor
that mathematics has somehow been proved inconsistent.The waves of such
disinformation can last for years or even decades.
*******************************************************************************
F. William Lawvere Mathematics Dept. SUNY
wlawvere@acsu.buffalo.edu 106 Diefendorf Hall
716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA
*******************************************************************************
Date: Wed, 7 Apr 1999 19:45:26 -0400 (EDT)
From: "R.A.G. Seely"
Subject: categories: Re: incompleteness of ZF
On Tue, 6 Apr 1999, F W Lawvere wrote:
> Using an old logician's trick (see eg Feferman on paths thru O, or even
> Goedel's original papers) as an
> April Fool joke
> may be amusing to some within the closed gates of a British University,
> but is irresponsible on the world network. Think of the hundreds of
> lurkers (who hesitate to speak up so that misconceptions can
> be discussed and clarified openly, but) who are now furthering the rumor
> that mathematics has somehow been proved inconsistent.The waves of such
> disinformation can last for years or even decades.
Curiously, my reaction to this has been rather different. We were
discussing Paul's note after the seminar here the other day, and apart
from one reply, I rather had the idea that most replies were aware
that this was a joke, but that it was a subtle one (not all that
subtle, perhaps, but a lot more subtle that what often passes as
humour on the net, for sure). And that finding the error was a
respectable response.
As for spreading disinformation, there has been no shortage of people
who look serious, (I avoid the harder question as to whether they are,
and what exactly that ought to mean) and who have been spreading tales
of the inconsistency of maths for decades. As a graduate student, I
often attended logic meetings where Edward Wette proved ever more
basic fragments of our subject inconsistent (I recall he got as far as
propositional logic, Peano arithmetic, and several branches of
physics. I am not sure he made any serious distinction between the
last case and the others.) Perhaps one point here is that anyone who
believes all he reads is a fool, and anyone who believes all he reads
on the net is a fool who cannot even learn from experience. (It takes
about 5 minutes to uncover patently silly things on the net! - not
even counting this one...)
So - I agree Bill raises a serious matter, but I think in this case it
may be an overreaction. Paul's note was certainly "fishy" (French
reference for those not in a French environment), but it was
essentially harmless. This group is comparitively closed (even your
average academic journal is probably more open). And if anyone missed
the joke up to now, surely that has been remedied. (If only in that
refutations appeared quickly.)
However, if I see a report on this on "60 minutes" I will eat my
hat...
- all the best, Robert
= rags =
=================================
From: Michael Abbott
Subject: categories: RE: incompleteness of ZF
Date: Thu, 8 Apr 1999 08:49:35 +0100
As a lurker who failed to either notice the date or understand any
detail of Paul's demonstration, I appreciate Lawvere's comments.
I do feel, though, that he is overstating the impact of Paul's little
jest. There were several immediate replies (to the effect, I think,
that F(lim X) is not lim F X), and no riposte from Paul. My own
reaction was: hmm, it'll be interesting if anything else comes out of
this.
I sympathise very much with Paul's "anti-ZF" stance; after all, hasn't
Lawvere set the basis for a non-set foundation of practical mathematics?
-----Original Message-----
From: cat-dist@mta.ca [mailto:cat-dist@mta.ca]On Behalf Of F W Lawvere
Sent: 06 April 1999 22:41
To: CATEGORIES@mta.ca
Subject: Re: categories: Re: incompleteness of ZF
Using an old logician's trick (see eg Feferman on paths thru O, or even
Goedel's original papers) as an
April Fool joke
may be amusing to some within the closed gates of a British University,
but is irresponsible on the world network. Think of the hundreds of
lurkers (who hesitate to speak up so that misconceptions can
be discussed and clarified openly, but) who are now furthering the rumor
that mathematics has somehow been proved inconsistent.The waves of such
disinformation can last for years or even decades.
************************************************************************
*******
F. William Lawvere Mathematics Dept. SUNY
wlawvere@acsu.buffalo.edu 106 Diefendorf Hall
716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA
************************************************************************
*******
Date: Thu, 08 Apr 1999 10:08:52 -0300
From: Robert Dawson
Subject: categories: April 1st & related matters
[Note from Moderator: With the posts just sent, this discussion should
come to a close. Readers of the list, and those posting to it, should be
aware that it contains about 500 addresses.]
*Had* Paul posted his clever hoax on, say, sci.math (which, AFAIK, he
did not) where it would be widely available to J. Random Lurker, I would
share Bill Lawvere's concerns. Had he placed it permanently on his web
server, with a big flashing link from his home page, advertised it by
spamming 100,000 random netizens, and issued a trilingual press release, I
would share those concerns to a much greater extent. However, CATEGORIES is
a mailing list; probably everybody who received Paul's posting also received
the two debunkings, Bill's comment, and is reading this even as, er, they
read this. I don't imagine that we *have* hundreds of lurkers. (Lurkers!
Please identify yourselves... I'm curious.)
While Bill is undoubtedly correct that not all readers of CATEGORIES
share his ability to look at Paul's joke and instantly recognize not only
the fallacy but its antecedents [I offer myself as a proof of the
nonemptiness of the complement], surely we all have been around the block
enough times to distinguish between a full formal proof and what Paul
presented? Even had it been in earnest, such an announcement would justify
only the reaction "Somebody thinks he's shown ... but I don't think he has
circulated a complete proof yet."
I suppose that it is possible that somebody, browsing at random, *might*
find it in the CATEGORIES archives, in years to come, and not read ahead to
find out what the world had had to say about this discovery back in the mad,
exciting days of the late C20. But anybody who could do this and not
realize that there was something odd going on would probably either (a) not
understand why anybody should care if ZF is consistent or not, or (b) have a
vague feeling that Goedel, or Escher, or somebody, already proved that, or
something, didn't he?
-Robert Dawson