FOM: Grothendieck universes; rampant f.o.m. amateurism
Stephen G Simpson
simpson at math.psu.edu
Thu Apr 15 18:24:30 EDT 1999
Colin McLarty 14 Apr 1999 17:35:35 writes:
> So, in ZFC, define a "universe" to be any set V(i) for i a limit
> ordinal greater than w. Thus each universe interprets Zermelo set
> theory with choice (and more as Kanovei pointed out).
No, no, no. What Kanovei pointed out (in 12 Apr 99 08:51:24) is that,
within ZFC, BY CHOOSING i APPROPRIATELY you can get V_i to have
additional properties: Sigma_n replacement for any fixed n, or full
replacement with domain restricted to sets of a fixed cardinality.
(This is a consequence of Levy reflection. Feferman's equiconsistency
result follows from this by a straightforward compactness argument.)
This is certainly NOT the case for arbitrary limit ordinals i > omega.
I didn't want to get into this discussion of Grothendieck universes,
but I can't resist commenting.
It's pretty clear to me that Grothendieck universes can be
straightforwardly eliminated wherever they have been cited in number
theory, and the number theorists who have cited them are well aware of
this. So the only right answer to Shipman's original question about
Grothendieck universes is a straightforward ``no''.
What's interesting to me is that there is so much confusion about this
relatively straightforward issue. But wait. Is it really only
confusion, or is it deliberate obfuscation? For the present, I will
be charitable and assume it's confusion, but I reserve judgment.
It seems to me that this kind of confusion (if that's what it is) has
to be the result of what I hereby dub rampant f.o.m. amateurism. In
other words, it's the kind of thing one would expect to happen when
arrogant f.o.m. amateurs try to practice professional f.o.m. ``without
a license'' as it were, and make a mess of it.
In the case of Grothendieck universes, arrogant f.o.m. amateurs may
conflate convenience or expediency with necessity. They may not be
able to distinguish ``Grothendieck universes are convenient'' from
``Grothendieck universes are needed''. Their error may flow from
inadequate understanding or appreciation of a key f.o.m. issue:
logical strength. When arrogant f.o.m. amateurs propagate such
errors, they may confuse colleagues who are not arrogant but merely
Unfortunately, the ranks of academia (mathematicians and others)
include quite a number of arrogant f.o.m. amateurs. These people
understand almost nothing of f.o.m., but they think they understand
everything, and that makes them dangerous. They may be competent or
even outstanding within their respective specialties, but when it
comes to f.o.m., they are a menace. They don't even know what the
right questions are. They can't grasp what f.o.m. professionals
patiently explain to them. They are very vocal. Wherever they go,
What can be done about this problem? The only remedy that I can see
is for arrogant f.o.m. amateurs to be made aware of their limitations.
They need to be told that f.o.m. is a deep and substantial subject,
different from mathematics itself. They need to be made aware that
the goals and methods of f.o.m. are different from the goals and
methods of specialized mathematical disciplines such as algebraic
topology and algebraic geometry.
Somebody needs to educate the arrogant f.o.m. amateurs. Who is
equipped to carry out this task? I think it's up to the
f.o.m. professionals. Let's think about how to do this. Perhaps the
FOM list can play a role in this educational project.
For a start, let's try to compile a catalog of arrogant
f.o.m. amateurism. Reuben Hersh comes to mind. The Bourbaki
phenomenon provides some dramatic examples. (Cf. ``the ignorance of
Bourbaki''.) Other instances have occurred or been reported on the
FOM list. A particularly good one is Harvey's story of a
mathematician who said that the independence of CH is of no interest
because second-order logic decides the question. Would anyone else
care to share some tales from the front?
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