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I would like to ask what the specific novel model-theoretic (or set-theoretic) techniques, if any, are that Mochizuki uses in his recent series of four papers. Section 3 of Inter-universal Teichm├╝ller Theory IV, as well as the comments on page 21 of paper I, seem to suggest that there are model-theoretic techniques in play; I don't, unfortunately, have the background in algebraic geometry to understand the presentation.

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It is my impression that model theorists have not really been much interested in applying the basic philosophy of model theory (i.e. that it is profitable to study a set of formulas by looking at a whole bunch of models as opposed to just a 'canonical' one) to abstract schemes. This kinda makes sense, since in one way the point of schemes is that they let you do geometry without actually having to worry about specific equations, and so they don't a priori seem to lend themselves well to model theory, which absolutely requires explicit formulas. Being mathematical objects however...(cont.) – KristianJS Sep 17 '12 at 13:58
they should, theoretically, be described by a formula in the language of set theory. The philosophy of model theory tells us it should be profitable to study this formula in lots of different models of ZFC (or, as I believe Mochizuki does, in ZFC + Grothendieck's Universe Axiom). It seems to me that the novelty is actually believing you can extract something even remotely concrete out of this! Now if only I actually understood anything about how he does it, then I could maybe submit an answer and not a comment.... – KristianJS Sep 17 '12 at 13:59
Oh, I should add that I'm pretty sure he's doing the above not just for schemes, but for even more abstract stuff (what he calls 'species' I think?) – KristianJS Sep 17 '12 at 14:04
It's not clear that Mochizuki actually uses the material in IUTT IV section 3 anywhere else in the papers. The only time he mentions 'species' or 'mutations' outside of that section are in reference to it in describing the contents of the papers. I'm leaning toward thinking that the material is not necessary, or at least not as earth-shattering as it seems. A standard result in topos theory is that one can talk about local ring objects in a topos, and there is a universal topos containing a local ring. Local rings are basic ingredients in scheme theory. What Mochizuki is doing looks like a ... – David Roberts Sep 18 '12 at 0:26
...Diophantine analogue, talking about models of the universal topos containing such-and-such a number-theoretic object. But this is just my impression from a category-theoretic point of view, I certainly don't know what he's doing on the number theory/Diophantine geometry/Teichmuller theory side. – David Roberts Sep 18 '12 at 0:28

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