universe
(in category theory/type theory/computer science)
of all homotopy types
type of types, object classifier,
codomain fibration
of homotopy n-types
of 0-truncated types/h-sets
of (-1)-truncated types/h-propositions
foundations
mathematical logic
first-order logic
type theory, homotopy type theory
set theory
material set theory
structural set theory
foundational axiom
basic constructions:
material axioms:
structural axioms:
axioms of choice:
Whitehead's principle
large cardinal axioms:
strong axioms
further
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Solomon Feferman, Typical ambiguity: Trying to have your cake and eat it too
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