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Paul Taylor | 1 Apr 1999 14:10
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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.
(Continue reading)

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Dusko Pavlovic | 1 Apr 1999 22:23
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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)  -->...
(Continue reading)

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Mike Oliver | 1 Apr 1999 17:52

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
(Continue reading)

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Thomas Streicher | 3 Apr 1999 17:12
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Re: incompleteness of ZF

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
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Phil Scott | 5 Apr 1999 19:25
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Position at Ottawa U.


The Department of Mathematics and Statistics of the
University of Ottawa is expecting one, if not several,
temporary positions for the upcoming year. The most
significant is the Canadian Mathematical Society
Fellowship.  This is a 3 year position (1 + renewable
for 2 years), at the level of Assistant Professor, 
with a reduced teaching load of 1 course per semester 
available July 1.   This  position recently became 
available when the current holder moved elsewhere. There 
may also be one or more temporary positions pending 
budgetary approval. 

We are hopeful to attract candidates interested in 
one or more of the following fields: 
	
	1. Categorical and Linear Logic
	2. Theoretical Computer Science
        3. Tensored Categories

We are also members of the Category Theory Centre in
Montreal (Team members and colleagues include: M. Barr,
M. Bunge, T. Fox, J. Lambek, M. Makkai, P. Panangaden, G.
Reyes, R. Seely).  The category team has weekly seminars 
(Ottawa is 2 hours from Montreal). We shall also be having  
frequent seminars of our team here at U. Ottawa. 

If you are interested, please contact one of us by
email right away:
(Continue reading)

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F W Lawvere | 6 Apr 1999 23:41
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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 <at> acsu.buffalo.edu               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

*******************************************************************************
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Michael Abbott | 8 Apr 1999 09:49
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RE: incompleteness of ZF

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 <at> mta.ca [mailto:cat-dist <at> mta.ca]On Behalf Of F W Lawvere
Sent: 06 April 1999 22:41
To: CATEGORIES <at> mta.ca
Subject: 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.

************************************************************************
*******
(Continue reading)

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Robert Dawson | 8 Apr 1999 15:08
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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
(Continue reading)

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R.A.G. Seely | 8 Apr 1999 01:45
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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
(Continue reading)

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F W Lawvere | 11 Apr 1999 19:51
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AMS Buffalo April 24/25, 1999

LAST ANNOUNCEMENT

               SMOOTH CATEGORIES IN GEOMETRY AND MECHANICS

          Special session of AMS Meeting No. 943 in Buffalo, N.Y.

                Diefendorf Hall, SUNY, Main Street Campus

                             April 24/25, 1999

note:  changes in Sunday schedule

SCHEDULE:  
Saturday, April 24, 1999
		
  9:00 - 9:20	James FARAN:
                        A Synthetic Approach to Characteristic Cohomology
                                   (Abstract # 943-18-72)
		
  9:30 - 9:50	Hirokazu NISHIMURA:		
                         Infinitesimal Calculus of Variations
                                   (Abstract # 943-18-71)

10:00 - 10:20	Jonathon FUNK:  
                        Lebesgue Toposes
                                   (Abstract # 943-18-65)

10:30 - 10:50	George JANELIDZE:  (joint work with Walter Tholen)
                        Strongly Separable Morphisms
                                   (Abstract # 943-18-73)
(Continue reading)

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