[FOM] Shinichi Mochizuki on set-theoretical/foundational issues
William Messing
messing at math.umn.edu
Sat May 25 09:03:59 EDT 2013
I think that Mochizuki is using the term set-theoretic universe in the
precise sense of a Grothendieck universe, as defined in SGA 4, Expose I
(appendix), that is in Bourbaki's set theory (as the appendix is written
by Bourbaki) which is ZF without the axiom of foundations, but with the
axiom of global choice, defined via Hilbert's epsilon symbol, called by
Boubaki \tau. Note explicitly that in expose I, ¶1, in addition to the
standard axiom, (UA): every set is an element of an universe, a second
axiom, (UB): if U is an universe, R{x} is a formula and there exists an
y\in U satisfying R{y}, then \tau_x(R{x}) is an element of U, is
assumed. This is used, among other things, to rigourously define the
Yoneda embedding of an U-category, C, into the category C^ of U-valued
presheaves on C.
William Messing
More information about the FOM
mailing list