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[4] Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations. PDF
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P67
Section 3: Inter-universal Formalism: the Language of Species
We shall refer to such models as ZFC-models. Recall that a (Grothendieck) universe V is a set satisfying the following axioms [cf. [McLn], p. 194]:
The various ZFC-models that we work with may be thought of as [but are not restricted to be!] the ZFC-models determined by various universes that are sets relative to some ambient ZFC-model which, in addition to the standard axioms of ZFC set theory, satisfies the following existence axiom [attributed to the “Grothendieck school” — cf. the discussion of [McLn], p. 193]: (†G) Given any set x, there exists a universe V such that x ∈ V. We shall refer to a ZFC-model that also satisfies this additional axiom of the Grothendieck school as a ZFCG-model. This existence axiom (†G) implies, in particular, that:
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