https://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf INTER-UNIVERSAL TEICHMULLER THEORY IV: LOG-VOLUME COMPUTATIONS AND SET-THEORETIC FOUNDATIONS Shinichi Mochizuki April 2020
P5 If, moreover, one thinks of Z as being constructed, in the usual way, via axiomatic set theory, then one may interpret the “absolute” - i.e., “tautologically unrelativizable” - nature of conventional scheme theory over Z at a purely settheoretic level. Indeed, from the point of view of the “∈-structure” of axiomatic set theory, there is no way to treat sets constructed at distinct levels of this ∈-structure as being on a par with one another. On the other hand, if one focuses not on the level of the ∈-structure to which a set belongs, but rather on species, then the notion of a species allows one to relate - i.e., to treat on a par with one another - objects belonging to the species that arise from sets constructed at distinct levels of the ∈-structure. That is to say, the notion of a species allows one to “simulate ∈-loops” without violating the axiom of foundation of axiomatic set theory - cf. the discussion of Remark 3.3.1, (i).
P68 On the other hand, by the axiom of foundation, there do not exist infinite descending chains of universes V0 V1 V2 V3 ... Vn ... - where n ranges over the natural numbers.
