Two Number Theory Items (and Woody Allen)

Here, I should mention that it’s not controversial that IUT uses an inter-universal setup to avoid set-theoretic contradictions (which Mochizuki calls “∈-loops”);

This part of Mochizuki’s claim always mystifies me. He claimed this back in the early 2000s. But it’s a complete non-starter. Number theory still works with the axiom of foundation removed. If you are somehow accidentally setting up your theory so that your mathematical objects (in this case, fields, topological groups, categories, etc) are in danger of defining a loop in the ∈-relation, then you are doing it wrong. This feels like Pierre Samuel’s book on algebra where the distinction between the natural numbers constructed via axioms of set theory and the natural numbers inside the reals is keenly felt, and so a “theorem” is proved that you can remove the subset of the reals and replace it by the original set of natural numbers, and get an isomorphic ring (and in fact this is done as a general result, and applied recursively from the complex numbers down through R, Q, Z and N). No one cares about this, and Mochizuki’s fears are even less well-founded (😏 ).

Also, the phrase “inter-universal” is meaningless unless you specify what “universe” is being referred to: actual Grothendieck universes? Something about toposes as places where you can do all of mathematics? Or the fluffy “universe” that is the metaphor for things like “alien ring structure”? I tried to make this point in this post where I quote Mochizuki using “universe” in a variety of conflicting ways.

Otherwise-contradictory relations between data imposed by certain links are o!set by assigning data to different universes with distinct labels,

The different labels thing is again a complete non-starter, and Mochizuki is so hung up on this it’s like an obsession. It’s like the issue with Samuel’s book mentioned above. It’s like worrying that you and I need to label our respective copies of the complex numbers with our names otherwise things might go wrong. Not to toot my own horn, but for people new to this, I also wrote something about this too. Now bringing in some kind of “different universes” machinery when one is literally trying to write down some (countable, even) indexing diagram for a functor is bananas. I don’t know if this is verbatim what Mochizuki told the author of the report, but it’s really not meaningful.

And this is just plain false:

ZFC doesn’t allow for Grothendieck universes, for Grothendieck universes imply a strongly inaccessible cardinal, which doesn’t exist in ZFC.

You cannot of course use ZFC to prove such cardinals exist, but you can’t prove they don’t exist in a model of ZFC.