[FOM] Shinichi Mochizuki on set-theoretical/foundational issues

Colin McLarty colin.mclarty at case.edu
Sat May 25 21:11:08 EDT 2013

It is clear he sometimes uses the term "universe" this way.  But are the
changes of universe  that he invokes, meant to be changes of this sense of
universe?  Yamashita says explicitly (in his Q2)  that as he sees it you
can think of "change of universe" either as changing Grothendieck universes
or as switching to a different scheme theory within one Grothendieck


On Sat, May 25, 2013 at 9:03 AM, William Messing <messing at math.umn.edu>wrote:

> 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
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