[FOM] Shinichi Mochizuki on set-theoretical/foundational issues
MartDowd at aol.com
MartDowd at aol.com
Sat Jun 1 10:37:30 EDT 2013
I've been looking further into the question of why Mochizuki wants to
extend the standard methods of category theory that have been in use by
mathematicians since the 1950's, and exactly what he is proposing. I'll
use the same bibliographic citations as in Mochizuki's "Inter-universal"
series of papers.
In IUTchIV Mochizuki states that:
On the other hand, if one restricts one's attention to such a category,
then one must keep in mind the fact that the structure of the category -
i.e., which consists only of a collection of objects and morphisms
satisfying certain properties! - does not include any mention of the
various sets and conditions satisfied by those sets that give rise to
the "type of mathematical object" under consideration. For instance, the
data consisting of the underlying set of a group, the group multiplication
law on the group, and the properties satisfied by this group multiplication
law cannot be recovered [at least in an a priori sense!] from the structure
of the "category of groups".
This much at least is subject to debate. A group is a tuple <G,1,x> where
1 \in G and x is a binary function on G, satisfying certain axioms. One
then defines a group homomorphism, and the category Grp whose objects are
the groups and whose morphisms from A to B are the pairs <f,B> where f is a
homomorphism with domain A and codomain contained in B. Showing that the
axioms of a category are satisfied makes use of the set-theoretic
definitions; proving properties of the category, such as completeness and
co-completeness, does also. In general, the "concrete" structure of Grp
may be referred to if necessary.
I'm going to leave off here for now; but I think that the question should
be
looked in to, whether Mochizuki's constructions can be carried out using
standard category-theoretic methods. In this case, the foundational issues
are as usual.
- Martin Dowd
In a message dated 5/31/2013 4:38:49 P.M. Pacific Daylight Time,
MartDowd at aol.com writes:
For any x\in V, P^-1(x) is a fiber, which is isomorphic to V as a
structure for the lnguage of set theory.
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