FOM: faking Grothendieck universes; VNBG; Hartshorne; Bourbaki; Johnstone; Conway

[Stephen G Simpson]

By VNBG I mean von Neumann Bernays G"odel set/class theory.

Colin Mclarty 23 Apr 1999 08:55:14 writes:

 > Hartshorne uses proper classes, so Simpson concludes that
 > Hartshorne's book can be formalized in VNGB. 

McLarty is both overstating and understating my remarks on VNBG in my
posting of 22 Apr 1999 17:42:45, entitled ``faking Grothendieck
universes''.  On the one hand, I only *suggested* that VNBG, perhaps
with global choice, may be the right place to formalize Hartshorne.  I
didn't *conclude* it.  On the other hand, I have additional reasons
for making that suggestion, beyond the mere fact that Hartshorne uses
proper classes.  The reasons are stated in my posting.

 > But Hartshorne also assumes that for any collections S and T there
 > is a collection of all "functions" from T to S.  VNGB supplies no
 > such collection when S and T are proper classes.

(Sigh.)  Where does Hartshorne say this?  Page reference, please.

It's true that Hartshorne *quantifies* over functors from S to T where
S and T are Abelian categories.  But that is perfectly OK in VNGB,
even when S and T are proper classes.  

I am very suspicious of McLarty's claim that Hartshorne assumes a
*collection* of all functions from S to T, when S and T are proper
classes.

 > You would have to extend VNGB by infinitely many higher
 > "types" above the type of classes. The natural way to do it is
 > by universes--the proper classes become universe sized sets.

I don't know of any f.o.m. professionals who consider this kind of
extension of VNGB particularly natural.  Moreover, nothing in
Hartshorne's book calls for such an extension.  Hartshorne never
mentions or assumes types above the type of classes, Grothendieck
universes, universe sized sets, or anything of the sort.

Once again, McLarty is blowing smoke, trying to misappropriate
Hartshorne's authority in order to puff up the importance of
Grothendieck universes.

Apropos my suggestion that VNBG might be the right place to formalize
the use of proper classes in category theory: It turns out that
Bourbaki said something similar in unpublished plans for the Bourbaki
set theory volume.  On page 379 of ``Modern Algebra and the Rise of
Mathematical Structures'' by Leo Corry, Bourbaki is quoted as saying:

  It was therefore decided that it will be better to enlarge the
  system in order to make room for categories.  At first sight,
  G"odel's system seems to be convenient. ...

(Bourbaki said this in 1956.  I assume Bourbaki was referring to VNBG
as expounded by G"odel in his 1940 monograph on the consistency of the
continuum hypothesis.  Corry doesn't comment on this issue.)

In the end Bourbaki decided not to bother with category theory,
presumably because categories were not useful enough in the branches
of mathematics that Bourbaki proposed to cover.  It might have been
different if Bourbaki had a volume on algebraic geometry.

Later I'll have more comments on the Leo Corry book, which I got from
the library.

Lawrence N. Stout 23 Apr 1999 09:50:49

 > It seems to me unlikely that one can settle questions in
 > foundations by looking in the index of standard texts.

Yes.  However, looking in the index of Hartshorne's text *did* help me
to dispose of McLarty's phony claims about Grothendieck universes in
Hartshorne's text.

 > Almost none will mention the rank of sets or the distinction
 > between sets and proper classes.  

Johnstone's book on topos theory is an exception to this rule.  It
explicitly distinguishes between ``small'' (i.e. set size) categories
and ``large'' (i.e. proper class size) categories.

 > Johnstone's book on Topos Theory doesn't have an entry in its index
 > for Grothendieck universes.

That's because the phrase ``Grothendieck universes'' does not appear
anywhere in Johnstone's book.  He says this explicitly on page xix of
his preface.  On the same page, he explicitly announces his intention
to be sloppy about set theory.  He justifies this by saying that he is
a fully-paid-up member of the Mathematician's Liberation Movement
founded by J. H. Conway, ``On Numbers and Games''.  On page xiii he
says that axiomatic set theory is not tremendously interesting.

In any case, Johnstone's subject index is very incomplete: only 4
pages, large type, and it doesn't contain entries for small or large
categories, although the phrase ``small category'' is used on page 1.

By contrast, Hartshorne's index is 19 pages, small type, and
apparently quite complete.  In every sense of the word, Hartshorne is
much more careful than Johnstone.  It's quite reasonable to assume
that, if Hartshorne had somehow used Grothendieck universes in his
text, then that would have been somehow reflected in his index.

-- Steve