Bourbaki and foundations

[John Baldwin]

Joe Shipman remarked May 12:    So far I have not seen anyone give an
explanation that makes sense of what is WRONG with set theory.

I have the same impression.  In writing this paper,  EXPLORING THE GENEROUS
ARENA
http://homepages.math.uic.edu/~jbaldwin/pub/Maddymay321.pdf

 for a volume in honor of Pen Maddy, I struggled with the same issue.  It
seems to me that the crucial issue is how to handle size. It is for this
reason that Mac Lane originially worked in Von Neumann-Godel- Bernays.


 Colin's recent post  pointed to Mac Lane's objections specifically to
ZFC.  My paper gives references to work by Mike Shulman which clearly lays
out a distinction between structural and material set theory.  But  he
argues that these are bi-interpretable theories.

However, Voevodsky raises different objections at
https://www.ias.edu/ideas/2014/voevodsky-origins

 "There were two main problems with the existing foundational systems,
which made them inadequate. Firstly, existing foundations of mathematics
were based on the languages of predicate logic and languages of this class
are too limited. Secondly, existing foundations could not be used to
directly express statements about such objects as, for example, the ones in
my work on 2-theories. "

I would appreciate anyone who can further clarify this objection.

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607
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