friedman: FOM: strike out?

Colin Mclarty cxm7 at
Fri Apr 23 08:55:14 EDT 1999

reply to Friedman and Simpson
>Well, it does at least appear that Simpson has struck out McLarty in 5:42PM
>4/22/99. The points Simpson is making would completely explain the
>experiences I reported in my earlier posting of 6:11PM 4/16/99:

	Simpson's points collapse.

	The technical one is the clearest. Hartshorne
uses proper classes, so Simpson concludes that Hartshorne's book 
can be formalized in VNGB. 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. 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.

	Simpson speculates that the terms "universe" and 
"Grothendieck topos" may never occur in Hartshorne. I am sure they
do not. Friedman stresses that his experts do not even know the
definition of "universe" or "topos", which may well be true.
Most of the experts are not very interested in the issue. 

	Algebraic geometry is like the French language in
this respect: you cannot understand it by picking out key words
you recognize. You must read entire sentences, even paragraphs.
And if you want to understand foundational issues in it you
must read them critically. 

	Simpson points out what obvious devices Hartshorne had
available to eliminate the proper classes. I entirely agree,
and point out that Hartshorne never uses nor even mentions these

	The rest of Simpson's post largely re-iterates familiar
ground. Most of the rest of Friedman's post quotes an earlier
post. I will say I do not doubt the quality of the experts. I 
doubt the value of studying foundations by playing "whisper down
the lane" with experts no matter how skilled.

	I will be gone over the weekend. If anything new comes up
on this thread I'll reply next week.

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