Subject: categories: Re: incompleteness of ZF

From: Thomas Streicher <[email-masked]>
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

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From: F W Lawvere <[email-masked]>
Date: Tue, 6 Apr 1999 17:41:04 -0400 (EDT)

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 
[email-masked]               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

*******************************************************************************

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From: Robert Dawson <[email-masked]>
Date: Thu, 08 Apr 1999 10:08:52 -0300

[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

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From: "R.A.G. Seely" <[email-masked]>
Date: Wed, 7 Apr 1999 19:45:26 -0400 (EDT)

> 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 =

=================================
<[email-masked]>
<http://www.math.mcgill.ca/~rags>